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right 
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mett 


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,          -  ..        .      _        .    \    . 

i:  ,.•* 

» 

* 

i 

6 

AN   ELEMENTARY  TREATISE 


ON 


RIGID  DYNAMICS 


•y 


Mm 


3! 


I 


« 


AN    ELEMENTARY  TREATISE 


ON 


RIGID   DYNAMICS 


BY 


W.   J.    LOUDON,    B.A. 

DKMONSIRATOR   IN   PHYSICS   IN   Till-    UNIVERSITY  OV  TORONTO 


V » 


*'  fxyjhcu  ayav 


Xclu  |[Jork 
MACMILLAN    AND    CO. 

AND    LONDON 
1896 

All  rights  reserved 


Copyright,  1895, 
Bv  MACMILLAN  AND  CO. 


NorhjooO  53rf3B 

J.  S.  CushiiiK  it  Co.  --  Berwick  &  Smith 

Norwood  Muss.  U.S.A. 


t 
ti 

t; 

tl 

Ci 

n 
k: 

C( 


i 


PREFACE. 


This  elementary  treatise  on  Rij;id  Dynamics  has  arisen 
out  of  a  course  of  lectures  delivered  by  me,  during  the  past 
few  years,  to  advanced  classes  in  the  University. 

It  is  intended  as  a  text-book  for  those  who,  having  already 
mastered  the  elements  of  the  Calculus  and  acquired  some 
familiarity  with  the  methods  of  Particle  Dynamics,  wish  to 
become  acquainted  with  the  ]:)rinciples  underlying  the  equations 
of  motion  of  a  solid  body. 

Although  indebted  to  the  exhaustive  works  of  Routh  and 
Price  for  many  suggestions  and  problems,  I  believe  that  the 
arrangement  of  the  work,  method  of  treatment,  and  more  par- 
ticularly the  illustrations,  are  entirely  new  and  original  ;  and 
that  they  will  not  only  aid  beginners  in  appreciating  fundamen- 
tal truths,  but  will  also  point  out  to  them  the  road  along  which 
they  must  travel  in  order  to  become  intimate  with  those  higher 
complex  motions  of  a  material  system  which  have  their  culmi- 
nating point  in  the  region  of  Physical  Astronomy. 

My  thanks  are  due  Mr.  J.  C.  Glashan  of  Ottawa,  who  has 
kindly  read  the  proofshects  and  supplied  me  with  a  large 
collection  of  miscellaneous  problems. 

W.  J.    LOUUOxN. 
University  ok  Toronto,  Aug.  19,  1895. 


^ 

1 


f 


CONTENTS. 


l\romcnts  of  Inertia 
Illustrative  Examples 


CHAPTER   I. 


PACK 
I 

II 


CHAPTER    II. 

Ellipsoids  of  Inertia 
Illustrative  Examples     . 

••••*» 

i6 

18 

Equimomental  Systems 
Principal  Axes 

20 
.         .         .       22 

Illustrative  Examples     . 

CHAPTER   III. 

.       26 

D'Alembert's  Principle  . 



•  31 

•  34 

•  35 

•  1>1 

•  39 

Impulsive  Equations  of  Motion 

Illustrative  Examples 

The  Principle  of  Energy 

Illustrative  Examples 

CHAPTER   IV 

Motion  about  a  Fixed  Axis.     Finite  Forces 
The  Pendulum       .... 
Illustrative  Examples     . 
Determination  of  .^  by  the  Pendulum 
Pressure  on  Fixed  Axis 
Illustrative  Examples     . 


45 
49 
51 
53 
58 
62 


Vll 


vUi 


CONTliNTS. 


CIIAITKK    V. 

Motion  about  a  Fixed  \xis.  IiniJiiisivc  Korcrs 
Centre  of  I'crciission  .... 
Illustrative  Examples  .... 
Initial  Motions.  Changes  of  Constraint 
liinstrativc  I')xanii)Ies  .... 
Tiic  liallislic  Pendulum 


PAOB 

70 

72 

74 
78 
78 
8S 


CHAI'IKR    VI. 

Motion  aljout  a  Fixed  Point.     Finite  Forces 
Anjj;ular  VY'locity  ....... 

Ciuneral  Equation.s  of  Motion  .... 

Equations  of  Motion  referred  to  Axes  fixed  in  Space 
luiler'.s  Equations  of  Motion  ..... 

Aufjular  Coordinate-,  of  the  IJody  .... 

Pressure  on  the  Fixed  Point  ..... 

Illustrative  Examples     ...... 

Top  spinning  on  a  Rough  Horizontal  Plane  . 

Top  spinning  with  Great  Velocity  on  a  Rough  Horizontal  Plane 

The  Ciyroscope  moving  in  a  Horizontal  Plane  about  a  Fixed  Point 


88 
88 

98 
100 

lOI 
IDS 

109 
III 

112 
120 
128 


CHAPTER   VII. 

Motion  about  a  Fixed  Point.     Impulsive  Forces 
Illustrative  Examples 


134 
137 


CHAPTER   VIII. 
Motion  about  a  Fixed  Point.     No  Forces  acting 


140 


CHAPTER   IX. 


Motion  of  a  Free  Body 
Illustrative  Examples 
Impulsive  Actions 
Illustrative  Examples 


145 

148 

156 
158 


CDNTKNTS. 


IX 


PAfJK 
70 
72 

74 
78 
78 

85 


CHAl'TKK   X. 

The  (iyroscope      ....... 

To  prove  the  Rotation  of  t lie  Karth  upon  its  Axis 
Hopkins'  i:irxtrical  Gyroscope       .         .         .         , 
Fcssel's  (iyroscf<pe        ...... 

(iyroscope  of  (Justav  Magnus         .         .        .         . 

NoTK  ON  Tin;  I'k.vih'LU.m  and  nii:    lOv    . 


•         *         t 


PACE 
162 
166 
168 

171 
172 


88 

88 

98 

loo 

lOI 

105 
109 
III 
112 
120 
128 


M ISCELLANEOUS   KXAMl'UliS 


•  • 


177 


137 


140 


145 
148 

156 
158 


RIGID    DYNAMICS. 


-•X »',  %^(>t>- 


CHAPTER   I. 


MOMENTS  OF    INERTIA. 


1.  In  attempting  to  solve  the  equations  of  motion  of  a  Rigid 
Body  in  a  manner  similar  to  that  employed  for  a  single  particle, 
it  will  be  found  that  certain  new  quantities  appear,  v  hich 
depend  on  the  extent  and  shape  of  the  body,  on  iis  density, 
and  on  the  way  in  which  it  may  be  moving  in  respect  of  some 
particular  line  or  system  of  coordinate  axes. 

2.  These  quantities  are  called  Moments  of  Inertia  and 
Products  of  Inertia.  A  moment  of  inertia  of  a  body  about  any 
line  is  defined  to  be  the  sum  of  the  products  of  all  the  material 
elements  of  the  body.by  the  squares  of  their  perpendicular  dis- 
tances from  this  line.  It  may  be  denoted  in  general  by  the 
letter  /,  and  when  /  is  expressed  in  the  form  AIK^,  where  M  is 
the  mass  of  the  body,  K  is  called  the  radius  of  gyratioti.  When 
the  body  is  referred  to  three  codrdinate  rectangular  axes,  the 
moments  of  inertia  about  the  three  axes  will  evidently  be 

A^^m^f^z^),    B=-.tm{a'^+x'^),    C=tm{x^+f), 

ni  being  the  mass  of  any  element  at  the  point  (,r,  y,  s),  and  the 
summation  being  taken  throughout  the  body. 

A  product  of  inertia  is  defined  with  reference  to  two  planes 
at  right  angles  to  one  another  and  is  found  by  multiplying  the 
elements  by  the  products  of  their  distances  from  these  coordi- 


In 


2  RIGID   DYNAMICS. 

nate  planes,  and  summing  them  throughout  the  body.     ProcKicts 
of  inertia  exist  in  sets  of  three,  and  for  three  rectangular  axes 

D=—in}>a,    E=^inax,    F=1nixy. 

3.  It  is  evident  that  when  the  law  of  ;//  is  known  and  the 
shajie  of  the  body  is  given,  the  finding  of  a  moment  or  of  a 
jjrcnluct  of  inertia  involves  an  integration  ;  and  the  following 
examples  will  serve  to  show  how  the  process  of  integration  may 
be  uf'id  for  this  purpose.  Further  on,  several  propositions  will 
be  given  by  which  the  method  may  be  usually  much  simplified. 

4.  Illustrations  of  finding  Moments  of  Inertia  by  Integration, 
(a)  A  uniform  rod  of  small  cross-section  about  a  line  perpen- 
dicular to  it  at  one  end. 

Here,  if  the  length  of  the  rod  be  2  a,  and  the  density  p, 


X2a 
px^d: 


'^dx--=M 


4a^ 


{b)  A  circular  arc  of  uniform  density  about  an  axis  through 
its  midpoint  perpendicular  to  its  plane. 


In  Fig.  I,  let  OA  =  r,  OCA  =  6,  OCB^n;  then  the  n.oment 
of  inertia  of  the  arc  BOD  about  an  axis  through  O  perpen- 
dicular to  the  plane  of  the  paper  is  2  Ipds  •  >^,  where  ds  is  an 
element  of  the  arc  at  A. 


1 


■.')v'^"^ 


■^$. 


MOMENTS   OF   INERTIA. 


kment 

Irpen- 

is  an 


1=8 pa^      sin2-,/0  =  4pr;3  p(i  -cos  0),/e  =  2Jl/(i  -''-— V. 
(c)  An  elliptic  plate,  of  small  thickness  and  uniform  density, 


Fig.  2. 

In  Fig.  2,  divide  the  plate  into  strips,  and  then  we  have 

/about  0V=4  Cpx^fdx=4p  C - x^^d^-,vdx=-.M--. 
c/"  Jo   a  4 

Similarly,  /about  OX=M — 

And  /  about  a  line  through  O  perpendicular  to  the  plate  will 
evidently  be  M 

For  a  circular  plate  a  =  b. 

(d)  A  rectangular  plate,  sides  2  a,  7  b. 

By  dividing  the  plate  into  strips  of  mass  m  it  will  be  seen  that 

./     4 /A  4<^ 

/about  side  2a^\\m \  =  M 


3  ' 

3  /  3' 

Also,  /  about  a  line  through  a  corner  perpendicular  to  the 
plate  is  M^irr-^b^).     For  a  square  plate  a  =  b. 


and  /aboutside2^  =  :iC;/^if^')==J/i^ 


4 


RIGID   DYNAMICS. 


{e)  A  triangular  plate. 

Let  the  triangle  be  ABC,  and  choosing  C  as  origin  of  coordi- 
nates, let  CA,  CB  be  the  axes.  Then,  dividing  the  triangle  into 
strips  parallel  to  AC^  an  elemental  mass  at  (,r,  y)  is  equal  to 
pdxdy  sin  C,  p  being  the  density.  The  distances  of  this  ele- 
ment from  AC,  BC,  and  the  point  C  are  xsinC,  ^  sin  C,  and 
V,f '^  +y^  +2xycosC. 

Hence  /  about  AC=  (     |         px^  sin^  Cd.rdy, 


°(a-x) 


/  about  BC=  I     i         py^  sin^  Cdxdy, 
and  /about  a  line  through  C perpendicular  to  the  triangle 


"(«      X) 


Xa    /*a 
I  p  sin  ^  (,r2  +7^  +  2  -ij  cos  C)dxdy. 

These  integrals  can  easily  be  evaluated,  and  the  moments  of 
inertia  expressed  in  terms  of  the  two  sides  and  included  angle. 

(/)  A  sphere  about  a  diameter. 

Dividing  the  sphere  up  into  small  circular  plates  of  thickness 


a. 
I 


m 


Fig.  3. 


MOMENTS   OF   INERTIA. 


dv,  as  in  Fi?^.  3,  wc  have 

I  about  a  diameter  =  2  i^p  •  irfdx  •  -  =p7r  i    (^i2_  ^-2^2^^^= j/  2  ^2 


1-2 


(^r)    A  right  circular  cone,  about  its  axis. 


Fig.  4. 

Dividing  the  cone  up  into  circular  strips,  perpendicular  to  its 
axis,  as  in  Fig.  4,  we  have,  if  a  be  its  height, 

I=^P'jTy^  •  dx    — ,  and  y=  --. 


1= 


irp 


d* 


-  Mlb^ 


M 


10 


5.  Products  of  inertia  can  be  evaluated  in  a  similar  way  ;  but 
as  they  are  generally  eliminated  from  the  equations  of  motion  by 
a  proper  choice  of  axes,  their  absolute  values  in  terms  of  known 
quantities  are  seldom  required. 

6.  Although  integration  gives  directly  the  values  of  moments 
and  products  of  inertia,  yet  the  process  becomes  tedious  for 
many  bodies  ;  and  the  following  propositions  will  be  found  use- 
ful for  their  determination,  when  one  knows  the  position  of  the 
centre  of  inertia. 


RIGID   DYNAMICS. 


Proposition  I.  —  To  connect  moments  and  products  of  inertia 
of  a  rigid  body  about  any  axes  ivitli  moments  and  products  of 
inertia  about  parallel  axes  through  the  centre  of  inertia. 


Fig.  5. 

Let  the  plane  of  Fig.  5  represent  any  plane  of  the  body  per- 
pendicular to  the  two  parallel  axes,  of  which  one  cuts  this 
plane  in  the  point  O,  and  the  other  passing  through  the  centre 
of  inertia  cuts  it  in  G.     Then  for  any  point  P  in  this  plane,  we 

have 

r^=:p'^+r'^  +  2P'  GM. 

Hence  for  the  whole  body  we  must  have 
tmr^=tm{p'^-\--t^'^  +  2p  •  GM) 

=  ^mp^+Smr'^-{-2p '^mGM 

=  Mp'^  +  tmr'\     since  tmGM^^o. 

Or,  as  it  may  be  written 

I=r,+Mp^ 

where  /  is  the  moment  of  inertia  about  any  axis,  and  /<,  is  that 
about  a  parallel  axis  through  the  centre  of  inertia,  and  /  is  the 
perpendicular  distance  between  ihe  axes. 

If  three  parallel  axes  be  taken  in  a  body,  of  which  the  third 
passes  through  the  centre  of  inertia,  and  a  plane  be  taken  cut- 


iiicts  of  inertia 
mi  products  of 
crtia. 


the  body  per- 

one  cuts   this 

ugh  the  centre 

this  plane,  we 


M=o. 


and  Iq  is  that 
\,  and  /  is  the 

?hich  the  third 
:  be  taken  cut- 


MOMKNTS   OF    INERTIA.  7 

ting  these  axes  perpendicularly  at  the  points  O,  O',  G,  then  we 
can  prove  for  the  whole  body,  as  before,  that 

/  about  axis  through  O  =r'-\-Ma\ 

/'  about  axis  through  0'  =  lo  +  Mb'^, 

OG=a,  0'G=b. 


and 
where 


If  G  happens  to  be  in  the  line  00\  this  relation  is  much  sim- 
plified.    Also,  if  OG  is  at  right  angles  to  00\  then 

which  relation  is  sometimes  useful  in  the  case  of  symmetrical 
bodies. 

It  is  evident,  moreover,  from  these  relations  that,  of  all  straight 
lines  having  a  given  direction  in  a  body,  the  least  moment^'of 
inertia  is  about  that  one  which  passes  through  the  centre  of 
inertia. 


Fig.  6, 


In  the  case  of  products  of  inertia,   similar   results   may  be 
obtamed.    Thus,  if  we  require  the  product  of  inertia  with  regard 


8 


RIGID   DYNAMICS. 


to  any  two  coordinate  planes  of  a  body,  let  parallel  planes  be 
taken  passing  through  the  centre  of  inertia.  Let  the  plane  of 
the  paper  in  Fig.  6  be  any  plane  of  the  body  perpendicular  to 
these  four  planes.  Then,  if  P  be  any  point  whose  coordinates 
referred  to  the  two  sets  are  {x\ y)  and  (-i',/'),  we  must  have  for 
the  whole  body 

Xmxy  =  tm  {x'  +/)  (/'  +  <]) 

=  ^vix'y  +f^inx'  +  q^mv'  4-  ^vipq. 

. :   %vix}'  =  %>fix'y  +  M  -  pq. 


Proposition  II. — ///  t/ic  case  of  a  lamina,  tJie  viomcnt  of 
inertia  about  any  axis  perpendicular  to  its  plane  is  cqnal  to  the 
sum  of  the  moments  about  any  two  perpendicular  lines  drazun  in 
the  plane  through  the  point  ivhere  the  axis  meets  the  lamina. 


For 


/=  ^m{x'^  +_;/2)  =  Imx"^  +  ^my"^. 


Proposition  III. —  To  find  the  moment  of  inertia  of  a  body 
about  any  line,  kmnving  the  moment  and  products  of  inertia 
about  any  three  rectangular  axes  drawn  through  some  point  on 
this  line. 

In  Fig.  7  let  the  three  rectangular  axes  be  OX,  OY,  OZ,  and 
let  P  be  any  point  of  the  body  {x,  y,  z),  and  ON  -diWy  line  drawn 
from  O,  inclined  at  angles  «,  /S,  7  to  the  axes. 

Then  /about  ON=tmPN'\  /W being  perpendicular  to  OY, 

and  /\V-  =  (9P2  _  OX^  =  (.1-2  +f  +  ,c'-^)  -  {x  cos  a  -\-y  cos  /3 -f .-  cos  7)2 

=  (x^  +J'2  +  ."2)  (cos^  a  +  cos^  ^  +  cos^  7)  —  (.r  cos  a 

+y  COS/3  +  .C'  cos  7)2 

=  (j'2  +  ,:;2)cos2  «+  •••  +  ••-  —  2J',CC0S /3cos  7 —  •••. 

.  •.  /=  "^M  l{y^  +  ^^)  cos^  a  H —  -!- 


J  —  2  Sm  \  yrj  cos  j3  cos  7 

+  •••  +  ••• 


MOMENTS   OK   INEKTIA. 


I  planes  be 
he  plane  of 
Midicular  to 
coordinates 
ist  have  for 


V 


moment  of 
qual  to  tJic 
s  drawn  in 
imina. 


I  of  a  body 

of  inertia 

le  point  on 

OZ,  and 
ine  drawn 

ar  to  OX, 
i  +  .:r  cos  7)2 

a 

1 4--  cos  7)2 


3s/3cos7 

••  +  ••• 


Fig.  7. 

=  A  cos^a  +  Z?  cos^/3  +  Ccos2  7  — 2  D  cos  /3cos7 

—  2E  cos  7  cos  «  —  2  /'cos  rt  cos  ^, 

A,  /),  C  being  moments  of  inertia  about  the  three  axes,  and 
D,  E,  F  products  of  inertia  with  regard  to  the  coordinate 
planes. 

In  this  expression,  it  will  be  seen  that  if  the  axes  of  coordi- 
nates be  so  chosen  that  D,  E,  I''  vanish,  then 

/=  A  cos^ a  +  B cos^ ^+C cos^ 7. 

Axes  for  which  this  holds  are  called  Principal  Axes,  and  A,  B,  C 
Principal  Moments  In  many  cases  such  axes  can  be  found  by 
inspection.  Thus,  if  a  body  be  a  lamina,  one  principal  axis  at 
any  point  is  the  perpendicular  at  that  point.  Also,  if  a  body  be 
one  of  revolution,  the  axis  of  revolution  must  be  a  principal 
axis  at  every  point  of  its  length.  And  it  may  be  stated  as  a 
general  rule  that  axes  of  symmetry  are  principal  axes. 


10 


RKilD    DYNAMICS. 


7.  In  most  of  the  problems  dealing  with  the  motion  of 
c'xtentled  bodies  the  axis  about  vvhieh  the  moment  of  inertia  is 
to  be  found  usually  passes  throujjjh  the  body  ;  but  it  is  apparent 
that  the  precedinj;-  propositions  apply  equally  to  all  cases  where 
the  axes  about  which  moments  of  inertia  are  required  do  not 
cut  the  body.  Thus  in  the  first  proposition  the  axes  parallel  to 
that  passing  through  the  centre  of  inertia  need  not  cut  the 
body  ;  in  the  case  of  a  lamina,  the  moment  of  inertia  about  any 
line  perpendicular  to  the  lamina  and  yet  not  intersecting  it  will 
still  be  the  sum  of  the  moments  about  any  two  perpendicular 
lines  drawn  at  the  point  where  the  axes  meet  the  plane  of  the 
lamina  produced  ;  and  similarly  the  moment  of  inertia  about 
any  line  outside  of  a  body  will  be  known  when  we  know,  at  any 
point  on  this  line,  the  moments  and  products  of  inertia  with 
respect  to  any  three  rectangular  axes  drawn  through  this  point. 


!  : 


;  I 


8.    TozvuscncV s  Theorem. 

A  closed  central  curve,  of  any  magnitude  and  form,  being 
supposed  to  revolve  round  an  arbitrary  axis  in  its  plane  not 
intersecting  its  circumference ;  the  moment  of  inertia  with 
respect  to  the  axis  of  revolution  of  the  solid  generated  by  its 
area  is  given  by  the  formula 

/=J/(rt2  +  3/,2), 

where  M  is  the  mass  of  the  solid  generated,  a  the  distance  of 
the  centre  of  the  generating  area  from  the  axis  of  revolution, 
and  Ji  the  arm  length  of  the  moment  of  inertia  of  the  area  with 
respect  to  a  parallel  axis  through  its  centre. 
For,  if  dA  be  an  element  of  generating  area, 

p  being  the  density,  and  x  a  variable  coordinate. 


MOMKNTS   OF    INKRTIA. 


II 


motion  of 
[  inertia  is 
s  apparent 
ises  where 
eel  do  not 
parallel  to 
3t  cut  the 
about  any 
tinj;  it  will 
■pendicular 
ane  of  the 
;rtia  about 
lovv,  at  any 
nertia  with 
this  point. 


5rm,  being 

plane  not 

lertia  with 

ited  by  its 


istance  of 

revolution, 

area  with 


Ikit,  by  the  symmetry  of  the  generating  area  with  respect  to  its 
centre,  i;(.iv//i)=o  and  i^(.rV//)=o. 

Illustrative  Examples  on  Miwients  of  Inertia. 

I.    Find  the  moment  of  a  rectangular  plate  about  a  diagonal, 
the  sides  being  2a,  2b. 

In  this,  applying  Pruj^osition  III.,  we  have 

/=/lcos2  6?  +  />'sin2^, 

the  centre  of  the  plate   being  the  origin,  and  A,  B  principal 

moments. 

3    aVf^ 


I=M 


Id'^l)^ 


2.  A  sphere  or  a  circular  plate,  about  a  tangent.  Apply 
Proposition  I. 

3.  Find  the  moments  of  inertia  of  a  rectangular  parallelo- 
piped  and  of  a  cube,  about  their  axes  of  symmetry  ;  also  about  a 
diagfonal. 


4.    The  moment  of  inertia  of  a  right  circular  cone  about  a 

s 


1 /;2    5^2  I  /,2 

slant  side  is  M „      „  ,  a  being  the  height  and  b  the  radiu 

20      a^  +  lP' 

of  the  base. 


5.  If  a  is  the  length  and  b  the   radius  of  a  right  circular 

cylinder,  the  moment  of  inertia  about  an  axis  through  the  cen- 

M  la'^       \ 
tre  of  inertia  perpendicular  to  its  axis  is  — ( — \-b^\. 

6.  The  moment  of  inertia  of  a  pendulum  bob,  density  p,  in 
the  form  of  an  cqui-convex  lens  of  thickness  2/ and  radius  a, 


ibout  its  axis  is  irp  I    {2ax—x'^)^dx. 


la 


ki(;nj  DYNAMICS. 


7.  Find  the  moment  of  inertia  of  an  anchor  ring  about  its 
axis. 

8.  The  moments  of  inertia  of  an  ellipsoid  about  its  three  axes 

arc  j\f — ^^^,   M — - — ,   M    - — 
5  5  5 

To  find  these,  either  divide  the  solid  ellipsoid  up  into  elliptic 
jilates,  or  deduce  from  the  case  of  a  sphere. 

9.  A  trian^^ular  plate  of  uniform  density. 

(I)  To  find  the  moment  of  inertia  about  the  side  HC. 


t       II 


l¥//////////////////^^^^^ 


JC 


a. 

Fig.  8. 


c 


In  Fig.  8,  divide  the  triangle  into  strips  of  mass  pydx,  where 
y=B'C. 

Then  /about  BC=^nydx  •  x'^=  p  \  ~ — '-ax^dx  =  M  -^  • 

'■^  '^Jo     p  6 

(2)  About  a  line  through  the  centre  of  inertia  parallel  to  BC. 

(3)  About  a  line  through  A  parallel  to  BC. 


MOMENTS  OF    INKRTIA. 


•3 


about  its 
three  axes 

to  elliptic 


?C. 


iv,  where 
2l  to  BC. 


(4)  About  a  median  lino. 

,/7 


In  Fig.  9,  divide  the  triangle  into  strips  parallel  to  BC,  as 
before,  and  \et  }'  =  B'C'.    Then  the  mass  of  a  strip  is  pj/t/.v  sin  JJ, 

and  its  moment  of  mertia  about  AD  is  pjv/.r  sin  D  •  -  sin''^  D. 

12 

Hence  /of  triangle  about  AD=— |     y^dx,  and  j'  =  — '^. 

rd'^  sin^  y; 


I=M- 


24 


./  ^2_|_^2 \  ^^  ^^  ^  being  the  three  sides. 


(5)  About  a  line  through  A,  perpendicular  to  the  plane  of  the 
triangle. 

Use  Fig.  9,  and  the  moment  of  inertia  will  be  found  to  be 

m    -  -2N 

4  \  3. 

(6)  About  a  line  through  the  centre  of  inertia,  perpendicular 
to  the  plane  of  the  triangle. 

36 

10.    Find  the  moment  of  inertia  of  a  hemisphere  about 
(i)   Its  axis. 

(2)  A  tangent  at  its  vertex. 

(3)  A  tangent  to  the  circumference  of  its  base. 

(4)  A  diameter  of  its  base. 


14 


RIGID   DYNAMICS. 


II.    The  moment  of  inertia  of  an  ellipsoidal  shell  of  mass  M 


fi^-c^ 


For  a  spherical  shell  about  a 


m 


about  the  major  axis  is  M 
diameter,  I=^M\a^.  ^ 

Deduce,  by  differentiation,  from  the  ellipsoid  and  the  sphere. 

12.  For  an  oblate  spheroid  (such  as  the  earth),  of  ex^en- 
tricity  e,  composed   of   similar  strata   of  varying  density,    he 

moment  of  inertia  about  its  polar  axis  is  fTrVi— e^j    p.-^dx, 

where  a  is  the  equatorial  radius  and  p  the  density  at  a  distance 
X  from  the  centre.  This  can  be  integrated  when  the  law  of  p 
is  known. 

13.  The   moment  of   inertia  of   a  paraboloid  of   revolution 

about  its  axis  of  figure  is  M  •  —,  where  r  is  the  radius  of  the 

■I 
base. 

14.  The  moment  of  inertia  of  the  parabolic  area  cut  off  by 
any  ordinate  distant  x  from  the  vertex  is  M^  x"^  about  the  tan- 

gent  at  the  vertex,  and  M  —  about  the  axis,  where  j/  is  the  ordi- 
nate corresponding  to  x. 

15.  The  radius  of  gyration  of  a  lamina  bounded  by  the  lem- 

niscate  r^^a^'  cos  26,  (i)  about  its  axis  is  -Vtt  — |;    (2)  about 

4 
a  line  in  the  plane  of  the  lamina  through  the  node  and  perpen- 
dicular to  the  axis  is  -Vtt  +  I  ;  (3)  about  a  tangent  at  the  node 

4 

IS   -Vtt. 


n 


ii; 


16.  To  find  the  radius  of  gyration  of  a  lamina  bounded  by  a 
parallelogram  about  an  axis  perpendicular  to  it  through  its  cen- 
tre of  inertia.     (Euler.) 

If  2a,  2  b,  be  the  lengths  of  two  adjacent  sides  of  the  parallel- 
ogram, then,  whatever  be  their  inclination, 

3 


MOMENTS   OF   INERTIA. 


'5 


17.  To  find  the  radius  of  gyration  of  a  hollow  sphere  about 
a  diameter.     (Euler.) 

5    a^  —  b^ 
a  and  b  being  the  external  and  internal  radii. 

18.  To  find  the  radius  of  gyration  of  a  truncated  cone  about 
its  axis.     (Euler.) 

3      a^-b^ 


10  = 


10     a^  —  b^ 


a,  b,  being  the  radii  of  its  ends. 

19.  The  moment  of  inertia  of  a  lamina  bounded  by  a  regular 
polygon  of  n  sides,  each  of  length  2  a,  about  an  axis  through  its 
centre  perpendicular  to  its  plane  is 


f(.  +  3co.^) 


And  from  this  it  can  be  seen  that  the  moment  of  inertia  about 
any  line  in  the  plane  of  the  lamina  through  the  centre 

12   \         ^  ;// 

20.  A  quantity  of  matter  is  distributed  over  the  surface  of  a 
sphere  of  radius  a,  so  that  the  density  at  any  point  varies  in- 
versely as  the  cube  of  the  distance  from  a  point  inside  distant  b 
from  the  centre.  Find  the  moment  of  inertia  about  that  diame- 
ter which  passes  through  the  point  inside,  and  prove  that  the 
sum  of  the  principal  moments  there  is  equal  to  2  M  {a^  —  b'^). 

What  if  the  point  be  outside  .? 


:; 


CHAPTER   II. 

ELLIPSOIDS   OF   INERTIA   AND   PRINCIPAL   AXES. 

9.    Ellipsoids  of  Inertia, 

At  any  point  C?  in  a  rigid  body  let  there  be  taken  three 
rectangular  axes  OX,  OV,  OZ,  as  in  Fig.  10.  Describe  with  O 
as  centre  the  ellipsoid, 

Ax^  +  By^  +  C:.^  —  2  Dyz  -  2  Ezx  —  2  Fxy = c, 


z 


Fig,    10. 
16 


ELLIPSOIDS   OF   INERTIA  AND   PRINCIPAL   AXES. 


17 


three 
vith  O 


where  A,  B,  C,  D,  E^  F,  have  the  meanings  already  attached  to 
them,  and  are  positive.  Then,  if  OP  be  any  line  drawn  from 
O,  and  cutting  the  ellipsoid  in  the  point  P,  the  moment  of 
inertia  of  the  body  about  OP  is 

A  cos"  a  +  B  cos^  /9  +  ^'cos^  7  —  •  •  •  =  /, 

where  «,  /3,  7  are  the  angles  which  OP  makes  with  the  coordi- 
nate axes. 

But  if  X,  y,  z,  are  the  coordinates  of  the  point  /*,  and  if 
OP  =  r,  we  must  have,  since  the  point  is  on  the  ellipsoid. 

And  since  this  relation  is  true  for  any  position  of  OP,  we  see 
that  the  moment  of  inertia  about  any  line  drawn  from  O  will  be 
inversely  proportional  to  the  square  of  the  corresponding  radius 
vector  cut  off  by  the  ellipsoid.  Any  such  ellipsoid  is  called  a 
Momental  Ellipsoid. 

10.  If  we  refer  the  ellipsoid  to  its  axes  OA,  OB,  OC,  then 
D,  E,  F  disappear,  and  the  axes  of  the  ellipsoid  are  therefore 
what  we  have  defined  as  Principal  Axes. 

11.  It  is  evident  that  any  set  of  principal  axes  at  a  point 
might  be  found  in  the  foregoing  manner,  namely,  by  construct- 
ing a  momental  ellipsoid  at  the  point  in  question  and  trans- 
forming to  the  axes  of  figure,  which  would  therefore  give  the 
directions  of  the  principal  axes.  And  it  may  be  stated  also  that 
three  principal  axes  necessarily  exist  at  each  point  in  space  for 
a  rigid  body,  since  the  above  [)rocess  can  always  be  performed. 

12.  From  the  properties  of  the  momental  ellipsoid  it  follows 
that  at  any  point  there  is,  in  general,  a  line  of  greatest  moment 
and  also  one  of  least  moment ;  if  the  ellipsoid  degenerates  into 
a  spheroid,  the  moments  of  inertia  about  all  diameters  perpen- 
dicular to  the  axis  of  the  spheroid  are  the  same  ;  if  it  becomes 
a  sphere,  as  in  the  case  of  all  regular  solids  at  their  centres, 
the  moments  of  inertia  about  all  lines  through  the  centre  are 


vmtm 


i8 


RIGID   DYNAMICS. 


ill 


1; 


equal,  a  proposition  which  can  be  applied  with  advantage  to 
the  cube,  proving  that  the  moments  of  inertia  about  all  lines 
through  the  centre  are  the  same. 

13.  For  a  lamina,  at  any  point,  the  section  made  by  the  cor- 
responding momental  ellipsoid  is  called  the  Momenta/  Ellipse  of 
the  point. 

Illtistrativf  Examples. 

1.  To  construct  a  momental  ellipsoid  at  one  of  the  corners  of 
a  cube. 

Taking  the  edges  as  axes,  A==B=C,  D=E=E,  and  the 
equation  for  the  momental  ellipsoid  becomes 

A  {x^  +  j'2  -I"  z^)  —  2  Dixy  -\-yz  +  ::-x)  =  c, 

which  on  transformation  would  give  a  spheroid  of  the  form 

and  it  can  be  seen  that  one  principal  axis  is  the  diagonal  through 
the  corner  in  question,  and  any  two  lines  at  right  angles  to  one 
another  and  to  the  diagonal  will  be  the  other  two  principal  axes. 

2.  To  find  the  momental  ellipsoid  at  a  point  on  the  edc,  j  of 
a  right  circular  cone. 

Choosing  axes  OX,  OV,  OZ,  as  in  Fig.  11,  it  is  evident  by 
inspection  that  D=^F=o,  and  the  axis  OY  is  one  principal  axis. 

Then,    if    AB  =  a,    OB  =  b\     BG^la,    and    A=m(^-+-\ 

*  \20        10^ 

B  =  A-\-Mb\   C=M^-^,  E  =  M—,   and  the  equation  of   the 

10  4 

momental  ellipsoid  at  O  is 

(3  <^H  2  rt2).i-2  -I-  (2 3  /;2  +  2  rt2)j2  ^  26  ^2-2  _  I o  abxz  =  c. 


t 
t 
r 
c 

V 

a 
r 


The  momental  ellipsoid  .,t  the  point  A,  or  at  any  point  along 
the  axis  AB.,  is  a  spheroid. 


of 


y2\ 


i 


ELLIPSOIDS   OF   INERTIA   AND   PRINCIPAL   AXES. 


19 


Fig.  11. 

3.  The  momental  ellipsoid  at  a  point  on  the  rim  of  a  hemi- 
sphere is 

4.  The  moi.iental  ellipsoid  at  the  centre  of  an  elliptic  plate  is 


5.    The  momental  ellipsoid  at  the  centre  of  a  solid  ellipsoid  is 

14.    The  Ellipsoid  of  Gyration. 

If  at  a  point  in  a  body  an  ellipsoid  be  constructed  such  that 
the  moment  of  inertia  about  any  perpendicular  drawn  from 
the  origin  on  a  tangent  plane  is  equal  to  Mp'^,  where  M  is  the 
mass  of  the  body  and  p  the  length  of  the  perpendicular,  it  is 
called  an  ellipsoid  of  gyration.  And,  since,  referred  to  its  axis, 
we  have  by  definition  A=Ahfi  about  the  axis  of  x,  B  =  Mb'^ 
about  the  axis  oi  y,  and  C=Me'^  about  the  axis  of  s,  its  equation 


must  be 


^-2 


1 


,2     „2 


.+  •'-  +  -  =  -—. 
ABC    M 


This  ellipsoid  may  also  be  used  to  indicate  the  directions  of 
the  principal  axes  ;   and,  from  the  form   of   its  equation,  it  is 


20 


RIGID   DYNAMICS. 


apparent  that  it  is  co-axial,  but  not  similarly  situated,  with  a 
momental  ellipsoid. 

15.  When  ellipsoids  are  constructed  at  the  centres  of  inertia, 
it  is  customary  to  speak  of  them  as  ccjitval  ellipsoids. 

16.  Equimonicntal  Systems. 

Two  systems  are  equimomental  when  their  moments  of  inertia 
about  all  lines  in  space  are  equal  each  to  each.  And  from  this 
definition,  taken  along  with  the  two  fundamental  proposition.s 
already  proved,  — 

/=  A  cos^ a -\- B  cos^ ^+C cos^ 7, 

it  follows  that  systems  will  be  equimomental  when  they  have 

1.  The  same  mass  and  centre  of  inertia. 

2.  The  same  principal  axes  at  the  centre  of  inertia. 

3.  The  same  principal  moments  at  the  centre  of  inertia. 

In  some  particular  cases  we  may,  instead  of  considering  a 
system  or  single  body,  use  a  simple  equimomental  system  in 
determining  its  motion  ;  but  generally  the  labour  of  proving  that 
systems  are  equimomental,  or  of  finding  a  simple  system  which 
will  be  equimomental  with  a  complicated  one,  is  greater  than 
that  of  solving  the  problem  directly.  The  following  examples, 
however,  will  serve  to  show  how  the  process  is  carried  out. 


> 


Illustrative  Examples. 


M 


I.    Show  that  three  masses,  each  equal  to — .placed  at  the 

3 

middle  points  of  the  sides  of  a  triangular  plate  of  mass  lil,  are 
equimomental  with  the  triangle. 

If  this  equimomental  system  be  assumed,  all  the  problems  in 
connection  with  a  triangular  plate,  such,  for  example,  as  finding 
moments  of  inerlia  about  the  sides,  perpendiculars,  and  median 
lines,  are  very  much  simplified  ;   but  the  difficulty  of  proving 


ELLIPSOIDS   OF   INERTIA   AND   PRINCIPAL  AXES.  2 1 

this  assumption  is  greater  than  that  of  solving  the  problenis,  as 
has  already  been  clone  by  a  direct  process. 

2.  In  an  elliptic  plate,  find  three  points  on  the  boundary  at 

which,  if  three  masses  each  equal  to  —  be  placed,  they  will  form 

3 
a  system  equimomental  with  the  plate,  whose  mass  is  M. 

3.  Show  that  three  points  can  always  be  found  in  a  plane  area 

AT 

of  mass  J/,  so  that  three  masses,  each  equal  to  — ,  placed  at 

3 
these  points  will  form  a  system  equimomental  with  the  area. 

The  situation  of  the  points  is  shown  in  Fig.  12,  which  repre- 
sents the  momental  ellipse  at  the  centre  of  iner'.ia  of  the  area. 
A  may  be  anywhere  on  the  boundary  of  the  ellipse ;  B  and  C 
are  so  situated  that  BD  =  DC  ?i\\(\  OD=DE. 


Fig.  12. 


4.    Find  the  momental  ellipse  at  the  centre  of  gravity  of  a 


triangular  area. 


22 


RIGID   DYNAMICS. 


5.  Th2  niomental  ellipse  at  an  angular  point  of  a  triangular 
area  touches  the  opposite  side  at  its  middle  point,  and  bisects 
the  adjacent  sides, 

17.    Pyincipal  Axes. 

To  find  the  principal  axes  at  any  point  of  a  rigid  body,  three 
rectangular  axes  might  be  chosen,  and  the  conditions  Xmxy  =  o, 
'^mj',a=o,  ^;«,av=o,  would  be  sufficient  to  solve  the  problem, 
cither  by  direct  analysis  or  by  the  construction  and  subsequent 
transformation  of  the  equation  of  the  momental  ellipsoid.  But 
this  process  would  often  be  tedious,  and  is  generally  unneces- 
sary. Usually,  by  inspection,  one  at  least  of  the  principal  axes 
can  be  found,  as  has  been  already  mentioned,  and  then  the  other 
two  may  be  obtained  by  the  following  propositions. 

Given  one  principal  axis  at  a  point,  to  find  the  other  two. 

Let  O  be  any  point  in  the  body,  and  let  OZ,  drawn  perpen- 
dicular to  the  plane  of  the  paper  be  one  perpendicular  axis. 
Take  any  two  lines,  OX,  O  Y,  at  right  angles  to  one  another  as 


■ 


Fig.  13. 


ELLIPSOIDS   OF   INERTIA   AND   PRINCIPAL  AXES. 


23 


axes  in  this  plane,  and  let  OA'',  OV  be  the  other  two  principal 
axes  at  O. 

Then  if  P  be  any  point  (x,  y)  or  {x'y'),  and  the  body  extends 
above  and  below  the  plane  of  the  paper,  we  must  have  as  a 
condition  that  OX',OY'  shall  be  principal  axes,  ^wx'y'  =  o 
throughout  the  body.  But  x'=x  cos  e-\-y  sin  6,  and  /  =  -x  sin  6 
+y  cos  6. 

Therefore  the  condition  becomes 

tm\  -x"^  sin  6  cos  O+y'^  sin  6  cos  6+xy  cos=^^-sin^  6]  =0, 
which  becomes,  on  reduction, 

tan  2  ^  =  — ---^Z_=^Z_, 
:Lnix'^-^iny'^     B-A 

according  to  our  previous  notation. 

If,  then,  A,  B,  F  be  found  in  respect  of  any  two  rectangular 
axes  OX,  OY,  6  is  known,  and  therefore  the  position  of  OX', 
OV. 


18.  The  condition  that  a  line  shall  be  a  principal  axis  at 
some  point  of  its  length  is,  that  taking  the  line  as  axis  of  s  and 
the  point  as  origin,  the  relations  'Ej;n's  =  o,  lmys  =  o  shall  be 
satisfied.  It  is  not  true,  however,  that  if  a  line  be  a  principal 
axis  at  one  point  of  its  length,  it  will  be  a  principal  axis  at  any 
other,  or  at  all  points  of  its  length.  For  example,  in  Fig.  1 1,  the 
line  OX  is  a  principal  axis  at  the  point  ^  on  account  of  the  sym- 
metry of  the  cone,  but  it  is  not  a  principal  axis  at  the  point  O. 
Similarly,  in  a  hemisphere,  any  diameter  of  the  base  is  a  prin- 
cipal axis  at  the  centre  of  the  base,  but  not  at  a  point  on  the 
rim.  There  is  one  case,  however,  in  which  a  line  is  a  principal 
axis  throughout  its  length,  and  as  this  is  of  some  importance, 
the  following  statement  and  simple  proof  are  given. 


19.    If  a  line  be  a  principal  axis  at  the  centre  of  inertia,  it 
will  be  a  principal  axis  at  every  point  of  its  length. 


H 


RIC.IU   DYNAMICS. 


Let  a  portion  of  the  body  be  represented  in  Fip;.  14,  O  being 

the  centre  of   inertia,  00'  the  principal  axis  at  the  centre  of 

inertia,  OA",  OY  any  rectani^ular  axes  at  O,  perpendicular  to 
00\  and  0X\  OV  parallel  axes  through  O'. 


Then  we  have,  by  a  previous  proposition : 

Imx'a' 3X.0'  =  t}nx:::xt  0-\-M{/ix), 
and  tmy's'  at  O'^^tmy:::  at  (9  +  M{hy). 

But  x=y  =  o  =  'lmxa  =  'ljiiy:;,  by  hypothesis. 

.•,  at  (9'  Smxy  =  o  =  -iny'c\ 

and  therefore  00'  is  a  principal  axis  at  O',  and  therefore  also 
at  any  point  in  its  length.  Conversely,  it  may  be  shown  that 
if  a  line  be  a  principal  axis  at  aP.  points  in  its  length,  it  must 
pass  through  the  centre  of  inertia. 


1 

1 


bcinpj 
trc  of 
lar  to 


ELLIPSOIDS   OF    INKKTIA    AND    PRINCIPAL   AXES.         2$ 

20.  To  dctorminc  the  locus  of  points  at  which  the  momcntal 
elhpsoid  for  a  given  body  de;;enerate.s  to  a  .spheroid,  and  the 
points,  if  such  exist,  at  which  it  becomes  a  sphere. 

Let  the  body  be  referred  to  its  principal  axes  at  the  centre 
of  inertia,  and  let  A,  />*,  and  C  be  its  principal  moments,  and 
J/ its  mass  :  — 

(i)  If  all  three  moments  be  unequal,  say  A  >  B  >  C,  there 
will  be  no  point  at  which  the  momental  ellipsoid  for  that 
body  will  be  a  sphere,  but  at  any  point  P  on  the  ellipse 

■»*JL_L     '■'    -J_      -n 
A-C^B-C~M'  '      ' 

or  on  the  hyperbola. 


A-B     B~C     J/' 


j  =  o. 


■e  also 
n  that 
;  must 


it  will  be  a  spheroid  with  axes  of  revolution  touching  the  conic 
at  P.  The  momental  ellipsoid  at  all  other  points  will  have 
three  unequal  axes. 

(2)  If  two  of  the  moments  be  equal,  and  each  less  than  the 
third,  say  A  >B=C,  there  will  be  two  points  at  which  the  mo- 
mental ellipsoid  for  that  body  will  be  a  sphere,  viz.,  the  points 

'A-C 


on  the  axis  of  .r,  distant  ± 


ve 


M 


from  the  centre  of  inertia. 


At  every  other  point  on  the  axis  of  x,  the  momental  ellipsoid 
will  be  a  spheroid  with  the  axis  of  x  as  axis  of  revolution.  At 
all  points  not  on  the  axis  of  x  the  momental  ellipsoid  will  have 
three  unequal  axes. 

(3)  If  two  of  the  moments  be  equal,  and  each  greater  than 
the  third,  sa.y  A  =  B>C,  the  momental  ellipsoid  for  that  body 
will  be  a  spheroid  at  every  point  on  the  axis  of  c,  or  on  the 
circle, 


.H/=^^, 


^=0. 


At  all  other  points  it  will  have  three  unequal  axes. 


26 


KIGID    DYNAMICS. 


(4)  It  /]  =  /)'=  6^,  the:  momcntiil  ellipsoid  will  be  a  sphere  at 
the  centre  of  inertia  and  a  spheroid  at  every  other  point. 

From  the  above  it  is  seen  at  once  that  in  the  majority  of 
bodies  there  is  no  point  for  which  all  axes  drawn  through 
it  are  principal  axes. 

Illustrative  Examples. 

I.  To  find  the  principal  axes  of  a  triangular  lamina,  at  an 
angular  point. 

One  principal  axis  is  the  line  drawn  through  the  angular 
point  perpendicular  to  the  lamina,  and  the  other  two  are  found 
in  the  following  way.  In  Fig.  15,  let  OA,  OB  be  two  rectan- 
gular axes,  OX,  OY  the  principal  axes  in  the  plane  of  the 
lamina.     Then  the  angle  which  OX  makes  with   OA   will  be 

2  F 
given  by  the  formula  tan  2  ^=  ^  —,»  where 


Fig.  15. 


^  =  moment  of  inertia  about  OA 


i 


' 


,.„(«  *) 


=JoU"       /'sin'^y^'''^v/j, 


-Jl 


and 


KLLIi'SUlDS   OK    INERTIA    AND   PRINCIPAL   AXLS.         27 

/?=  moment  of  inertia  about  t?Z? 
» 
\^  y^         /3  s I  n  (0  (.1-  +_;/  cos  (afdxdy, 


[  y^         fJsina)(.r+jcosa))>'sin  W.tv/;/, 


p  being:  the  density  of  the  lamina,  and  the  axes  of  x  and  y  lying 
along  the  sides  of  the  triangle. 

It  will  be  found,  on  evaluating  these  integrals,  that 

tan  J  ^ -    ^  ''^'"  *"  ^'^  '^"^  ^"^  "^ 
d^  +  r?/^  cos  tu  4-  b^  cos  2  w 

As  a  simple  case,  let  w  =  ^;  then  the  triangle  is  right  angled, 
and  tan  2  ^  =  -^— _,  as  can  easily  be  found  independently  of  the 
above  formula. 

2.  To  find  the  principal  axes  at  any  point  of  an  elliptic 
lamina. 

In  Fig.  16,  let  O'  be  the  point  (a,  /S)  at  which  we  require  the 


"! 


^rl 


»n 


Fig.  16. 


28 


RIGID   DYNAMICS. 


; 


i     I 


principal  axes.     Then  the  angle  6  which  O'X'  makes  with  the 
principal  axis  at  O'  is  given  by 


tan  2  0- 


2F 


B-A 


where 


A  =  I  about  0'X'  =  I  about  QX^  M^^ 
^  =  / about  C>' 7'=/ about  OY+Ma^ 


:^g  +  ,4 


and 


F=  tmxy  =  %7nxy  +  Ma^  =  Ma^. 

2Ma^ 


.'.  tan  2  6= 


^  8«/3 

(rt2-^2)+4(«2_^2)- 

The  third  principal  axis  is,  of  course,  at  right  angles  to  the 
lamina,  through  the  point  O'. 

3.  To  find  at  what  point  a  side  of  a  triangle  is  a  principal 
axis. 

Fig.  17  shows  the  construction  and  proof.  BC  is  the  side  in 
question,  and  is  bisected  at  B.  AD  is  drawn  perpendicular  to 
BC,  and  DE  is  bisected  at  O.     Then,  taking  the  equimomental 

svif-em  —  at  the  middle  points  of  the  sides,  in  order  that  the 
.        .      3 

inertia-pri:)duct  F  may  vanish,  the  principal  axis  perpendicu- 
lar to  the  side  BC  must  bi.sect  the  join  of  the  mid-points  of  the 
sides  AB  and  AC,  hence  BC,  OY  are  the  principal  axes  in  the 
plane  of  the  lamina  at  the  point  O. 


ELLIPSOIDS   OF   INERTIA   AND   PRINCIPAL   AXES.         29 


4.  Find  the  principal  axes  at  any  point  of  a  square  or  a  rec- 
tangular plate. 

5.  Find  the  principal  axes  at  any  point  within  a  cube  or 
a  rectangular  parallelopiped. 

6.  The  principal  axes  at  any  point  on  the  edge  of  a  hemi- 
sphere are,  one  touching  the  circumference  of  its  base,  and  two 
others,  given  by  the  relation  tan  2  ^  =  |. 


7.  The  principal  axes  at  any  point  on  the  edge  of  a  ri<yht 
circular  cone  are,  one  touching  the  circumference  of  its  base, 
and  two  others,  given  by  the  relation  tan  2  0= — 19JI^  where  ^ 

is  the  altitude  of  the  cone  and  b  the  radius  of  the  base. 

If  rt  =  2<^,  then  one  of  the  principal  axes  passes  through  the 
centre  of  inertia,  and  at  the  centre  of  inertia  itself  all  axes  are 
principal  axes. 

8.    Find  the  principal  axes  at  any  point  in  a  lamina  in  the 
form  of  a  quadrant  of  an  ellipse. 


m 


\\ 


'M  M 


Hi  i\ 


HI 


J ' 


ii 


30 


RIGID   DYNAMICS. 


9.  Determine  the  condition  that  the  edge  of  any  tetrahe- 
dron may  be  a  principal  axis  at  some  point  of  its  length,  and 
find  the  point. 

10.  Two  points  P  and  Q  are  so  situated  that  a  principal 
axis  at  P  intersects  a  principal  axis  at  Q.  Then  if  two  planes 
be  drawn  at  P  and  Q  perpendicular  to  these  principal  axes, 
their  intersection  will  be  a  principal  axis  at  the  point  where  it  is 
cut  by  the  plane  containing  the  principal  axes  at  P  and  Q. 

(Townsend.) 

21.    In  determining  the  directions  of  the  principal  axes  by  aid 


of  the  relation  tan  2^  = 


2F 


B-A 


,  if  F=o  and  at  the  same  time 


B  —  A,  then  the  value  of  6  is  indeterminate,  and  any  two  axes 
perpendicular  to  the  given  one  and  to  one  another  are  principal 


77 


axes  ;  if  B  —  A  and/'is  finite,  then  tan  2  ^  =  infinite  and  2  6=~  ; 

2 


TT 


if  F=Oy  and  B  is  not  equal  to  A,  then  tan  26  =  0  and  ^  =  0  or  — 

2 


M 

i 


nil 


tetrahe- 
igth,  and 


principal 
o  planes 
)al  axes, 
lere  it  is 

Q. 
vnsend.) 

;s  by  aid 
ne  time 

wo  axes 
irincipal 


=oor^. 


CHAPTER    III. 

D-ALEMBERT'S   PRINCIPLE. 

22  In  determining  the  motion  of  a  single  particle  of  mass 
;//,  three  rectangular  axes  are  chosen,  and  \i  X,  Y  Z  be  the 
accelerations  in  the  directions  of  these  three  axes,  the  equation 
of  the  path  is  found  from  the  relations 


;;/ 


mV, 


d^^ 
dt^ 

In  the  case  of  a  body  which  is  composed  of  a  number  of  parti- 
cles collected  into  what  is  termed  a  rigid  body,  if  we  follow  the 
above  method,  we  get  three  relations  of  the  type 

d\x         ^     , 

where  /^  arises  from  the  internal  molecular  actions,  and  X  is 
as  before,  the  acceleration  of  a  single  particle  whose  mass  is  nl 
For  every  particle  we  should  get  similar  relations,  the  value  of 
/i,  however,  changing  from  point  to  point  in  the  body.  We  can 
proceed  no  further  in  the  solution  of  such  equations,  owin-  to 
our  imperfect  knowledge  of  the  value  and  variation  of  /,  '^But 
the  Principle  of  D'Alembert  enables  us  to  form  an  equation 
independent  of  the  internal    molecular  actions    by  taking   the 

3J 


I,; 

III 


r:! 


Ui 


II 


■i 


32 


RIGID   DYNAMICS. 


sum  of  all  the  forces  acting  on  the  individual  particles  which 
compose  the  body.     Thus,  for  all  the  particles,  we  must  have 


-f''^'^')  =  -(^'^^^')+-(/i)' 


df 


t{in'^~^=^t{mY)  +  t{f^U 


[in'^^  =  t{mZ)+1{f^), 


and  D'Alembert's  principle  states  that 

23.    The  equations  of  motion  of  a  rigid  body,  then,  are 


(A) 


d'^v 
Each  force  of  the  type  in  — ^  is  termed  an  effective  force  ;  and 

the  above  relations  are  equivalent  to  saying  that  the  effective 
forces,  if  reversed,  would  be  in  equilibrium,  with  the  external  or 
impressed  forces  ;  they  may  be  looked  upon  either  as  equations 
of  motion  or  as  conditions  for  equilibrium. 

24.  It  is  evident,  also,  from  this  same  principle,  that  if  we  take 
the  S7im  of  all  the  moments  of  the  effective  forces,  these,  if  re- 
versed, will  balance  the  sum  of  all  the  moments  of  the  external 
forces.  Consequently,  for  any  set  of  rectangular  axes,  we  must 
also  have 


(A) 


I  ! 

\ 


D'ALEMBERT'S   PRINCIPLE. 


33 


2 


wlj 


>  — —  —  r:  — ^ 


dfi 


dfi)_ 
dh 


,     dlv 

dfi        df' 


7)l[ 


=  N, 


(B) 


V    dfi     -^  dfl)_ 
where  L,  M,  iV^are  the  couples  produced  by  the  external  forces. 

25.  It  may  be  stated  b-re  that  D'Alembert's  principle  holds 
also  in  the  case  of  a  system  of  bodies  moving  under  their  mutual 
actions  and  reactions,  and  applies  to  the  motion  of  liqp.ids.  It 
is  a  direct  consequence  of  Newton's  Third  Law  of  Motion. 

26.  Deductions  from  D'Alembert's  Principle. 
Taking  any  one  of  the  equations  (A),  we  have 

But,  by  definition  of  the  centre  of  inertia, 

'^mx=Mx, 


and 


2;/.^  =  J/^'-^' 


dt"^      '"  df^ 
Therefore  the  above  relation  becomes 

M 


dt^ 


and  similarly  for  the  other  two. 

(i)  Hence,  the  motion  of  the  centre  of  gravity  of  a  sy  stein  under 
the  action  of  any  forces  is  the  same  as  if  all  the  mass  tvere  col- 
lected at  the  centre  of  inertia  and  all  the  forces  were  applied  there 
parallel  to  their  former  direction. 

And  so  the  problem  of  finding  the  motion  of  the  centre  of 
inertia  of  a  system,  however  complex,  is  reduced  to  finding  that 
of  a  single  particle. 


ill 


'Ak 


4 


v:\\ 


'i'f 


■if 


•■( 


\k 


34  RIGID   DYNAMICS. 

Moreover,  taking-  one  of  the  equations  (B), 


siii^c  we  may  choose  the  origin  of  coordinates  at  any  point,  let 
it  be  :.D  chosen  that  at  the  time  of  forming  these  equations  the 
centre  of  inertia  is  coincident  with  it,  but  moving  with  a  certain 
velocity  and  acceleration.  Then,  evidently,  we  must  obtain  a 
relation  of  the  same  form  as  the  foregoing,  just  as  if  we  had 
considered  the  centre  of  inertia  as  a  fixed  point.  In  other 
words,  such  a  relation  as  the  above  will  hold  at  each  instant  of 
the  body's  motion,  independently  of  the  origin  and  of  the  posi- 
tion of  the  body. 

(2)  Hence,  the  motion  of  a  body,  under  the  action  of  any  finite 
forces,  about  its  centre  of  inertia,  is  the  came  as  if  the  centre  of 
itiertia  were  fixed  and  the  same  forces  tvere  acting  on  the  body. 

27.  The  two  previous  deductions  are  known  as  the  principles 
of  the  Conservation  of  the  motions  of  Translation  and  Rotation, 
and  show  us  that  we  may  consider  the  two  motions  indepen- 
dently of  one  another. 

28.  Impulsive  Equations  of  Motion. 

Since  an  impulse  can  be  measured  only  by  the  change  of 
momentum  it  induces  in  a  body,  in  applying  D'Alembert's  Prin- 
ciple to  impulsive  forces  we  must  alter  the  expressions  for  the 
effective  forces,  which  will  be  represented  not  by  the  products 
of  masses  and  accelerations,  but  by  the  products  of  masses 
and  changes  of  velocity.  All  the  preceding  relations  will  hold 
equally  for  impulsive  forces  if  we  then  write  changes  of  velocity 
for  accelerations. 

Thus,  such  a  relation  as 

2w  — '-  =  "^mX, 
dt^ 


\ 


D'ALEMBERT'S   PRINCIPLE. 


35 


for  finite  force:  will  become 


2;;/ 


fch\ 

Kit)  ~ 


dx 

dt 


=  1X, 


for  impulsive  forces  where  the  velocity  of  each  particle  of  m-^ss 


m  is  changed  from  -^-  abruptly  to  (  -^ 

dt  ^         \dt 


by  the  action  of  an 


impulse  X.  And  it  may  be  said,  generally,  that  equations  of 
motion  for  impulsive  forces  can  be  obtained  from  the  corre- 
sponding equations  for  finite  forces  by  substituting  in  the  latter 
changes  of  velocities  for  accelerations. 

29.  In  forming  any  relations  for  impulses,  it  must  be  borne 
in  mind  that  all  finite  actions,  such  as  that  of  gravity,  are  to  be 
neglected  ;  after  the  impulse  has  acted,  the  subsequent  motion 
will,  of  course,  be  found  by  applying  the  equations  for  the 
finite  forces  which  usually  are  called  into  play  after  the  impulse 
has  operated. 


''1 


I' , 


Illustrative  Examples. 

I.  A  rough  uniform  board,  of  length  2  a  and  mass  rn,  rests  on 
a  smooth  horizontal  plane.  A  man  of  mass  M  walks  from  one 
end  to  the  other.     Determine  the  motion. 

This  example  furnishes  an  excellent  illustration  of  the  truth 
of  D'Alembert's  principle,  which  asserts  that  the  motion  of  the 
centre  of  inertia  of  the  system  will  be  the  same  as  if  we  applied 
there  all  the  forces  external  to  the  system,  each  acting  in  its 
proper  direction.  All  the  forces  at  the  centre  of  inertia  are 
then  downwards,  and  as  the  centre  of  inertia  cannot  move 
downwards,  it  must  therefore  be  at  rest ;  and  as  the  man  walks 
along  the  whole  board,  he  will  therefore  advance  relatively  to 


the  fixed  horizontal  plane  through  a  distance 
board  will  recede  through  a  distance  ^^ 


2  ma 


,  and  the 


',\ 


*  ill 


36 


RIGID   DYNAMICS. 


Analytically,  we  have  for  the  motion  in  a  horizontal  direction, 
since  there  are  no  horizontal  forces  external  to  the  system,  the 
equation 

2;/^--— =0. 


-f- 


and 


(it 


=  o  or  constant. 


If  the  man  and  board  start  from  rest,  as  we  have  supposed, 
then 

^-_ 
dt' 


=  o. 


,".  .*•= constant. 


which  means  that  the  position  of  the  cen're  of  inertia  remains 
unaltered  throughout  the  motion  of  the  two  parts  of  the  system. 

2.  Two  persons,  A  and  B,  are  situated  on  a  smooth  horizon- 
tal plane  at  a  distance  a  from  each  other.  If  A  throws  a  ball 
to  B,  which  reaches  B  after  a  time  t,  show  that  A  will  begin  to 

slide  along  the  plane  with  a  velocity  — -,  where  M  is  his  own 
mass  and  in  that  of  the  ball. 


3.    A  person  is  placed  on  a  perfectly  smooth  surface, 
may  he  get  off } 


How 


4.  Explain  how  a  person  sitting  on  a  chair  is  able  to  move 
the  chair  along  the  ground  by  a  series  of  jerks  without  touching 
the  ground  with  his  feet. 

5.  How  is  a  person  able  to  increase  his  amplitude  in  swing- 
ing without  touching  the  ground  with  his  feet .-' 


;ction, 
n,  the 


)osed, 


lains 
:em, 

izon- 
ball 
n  to 

own 


D'ALK.MI'.KKT'S    I'RINCII'LE. 


37 


6.  Explain  dynamically  the  method  of  high  jumping  with 
a  pole;  and  show  that  a  man  should  be  able  to  jump  as  far 
on  a  horizontal  plane  without  a  pole  as  with  one. 

7.  Two  coins,  a  large  and  a  small  one,  arc  spun  together 
on  an  ordinary  table  about  an  axis  nearly  vertical.  Which  will 
come  to  rest  first,  and  why.'' 

8.  A  circular  board  is  placed  on  a  smooth  horizontal  plane, 
and  a  dog  runs  with  uniform  speed  around  on  the  board  close 
to  its  edge.     Find  the  motion  of  the  centn^    f  the  board. 

30.    TJic  Principle  of  Energy. 

Before  entering  upon  the  discussion  of  the  motion  of  a 
rigid  body,  what  is  known  as  the  principle  of  energy  will  be 
explained,  as  it  is  exceedingly  useful,  and  often  gives  a  partial 
solution  of  a  problem  without  any  reference  to  the  equations 
of  motion,  and  in  many  cases  furnishes  solutions  which  are 
both  simple  and  elegant  when  compared  with  those  obtained 
by  the  use  of  Cartesian  coordinates. 

If  a  single  particle  of  mass  ;;/  be  moving  along  the  axis  of  x, 
under  the  action  of  a  force  F  in  the  sa.  e  direction,  we  have, 
as  the  equation  of  motion, 

iPx     ^ 


I 'I 


'Mi 

'■'I 


1 

'A 


low 


lOve 


Ung 


And  multiplying  both  sides  by  ~  and  integrating,  we  get 


dx 


where  V  is  the  initial  value  of  v  or 


dt 


ng- 


The  expression  on  the  left-hand  side  of  the  equation  is  the 
change  in  kinetic  energy,  which  is  equal  to  the  zvork  done  by 
the  force  from  o  to  x. 


38 


kldlD   DYNAMICS. 


!l 


What  is  true  of  :i  sinj^le  force  acting  in  a  definite  direction 
and  of  a  single  jxirticl^'  of  mass  m  is  also  true  of  a  number 
of  forces  acting  on  a  rigid  body  or  on  a  system.  Then  the 
analytical  expression  for  the  work  done  by  a  system  of  forces 
becomes 


^;;/J  \Xiix  + )  'dy  +  Zih), 


which  must  be  equal  to 

In  the  general  case,  where  bodies  move  with  both  translation 
and  rotation,  the  total  kinetic  energy  can  easily  be  shown  to 
be  that  due  to  translation  of  the  whole  mass  collected  at  the 
centre  of  inertia,  together  with  that  due  to  rotation  about  the 
centre  of  inertia  considered  as  a  fixed  point. 

For  if  Xf  y,  a  be  the  coordinates  of  any  particle  of  mass  m 
and  velocity  v  at  time  t,  and  x,  y,  1:  be  the  codrdinates  of  the 
centre  of  inertia,  |,  ?;,  ^  the  coordinates  of  the  particle  referred 
to  the  centre  of  inertia,  then  the  total  kinetic  energy  is  equal  to 


^^vriP'  =\^in 


,A2 


1(f)- 


^^-m 


\ 


=  J5;. 


dt]      \dtJ 


'\ 


(f)i--' 


-'"Id 


+ 


IJHfJI' 


since  by  definition  of  the  centre   of   inertia  the  other  terms 
disappear.     This  proves  the  proposition. 


31.  According  to  the  kind  of  motion  and  the  choice  of  coor- 
dinates and  origin,  this  expression  for  energy  will  assume 
various  forms  which  will  be  given  under  the  discussions  of  the 
special  cases  throughout  the  treatise. 

Twice  the  energy  is  termed  the  vis  viva. 


D'ALEMMERT'S    PRINCIFLE. 


39 


32.  To  find  the  work  clone  by  an  impulse;  let  Q  be  the 
measure  of  an  impulse  which,  acting  on  a  particle  of  mass  ;// 
moving  with  velocity  F,  changes  its  velocity  suddenly  to  v ; 
then  the  kinetic  energy  is  changed  from  ,]  m  V^  to  \  mvK 

Work  done  by  the  impulse 

=  \Q'{v+V), 

since  the  impulse  is  measured  by  the  change  of  momentum  and 
Q  is  therefore  equal  to  mv  —  m  V. 

A  similar  relation  will  evidently  apply  to  a  rigid  body  where 
V  and  V  are  the  velocities  of  the  point  of  application  of  the 
impulse  resolved  in  the  direction  of  the  action  of  the  impulse. 

Illustmth'c  Examples  on  Energy. 

r.  A  rod  OA,  of  length  2.7,  fixed  ;'t  (9,  drops  from  a  horizon- 
tal position  under  the  action  of  gravity  :  find  its  angular  velocity 
when  it  is  in  the  vertical  position  OB.     (See  Fig.  18.) 


o 


a. 


,111 


1 1 

'■ii 


+ 


■.oi 


.1^ 


B 


Fig.   18. 


40 


lUC.lD    DYNAMICS. 


Here,  the  work  done  by  gravity  in  moving  the  rod  from  the 
position  OA  to  OB  is  Jlf^'-a,  M  being  the  mass  of  the  rod.  The 
rod  starts  from  rest  in  the  position  OA,  so  that  when  in  the 
position  OB  the  change  in  kinetic  energy  is  measured  simply 
by  the  energy  in  the  position  OB.  This  kinetic  energy  is  equal 
to  the  expression  .]  ^)m>^,  2>  being  the  velocity  of  any  particle 
m  ;  and  the  linear  velocity  of  any  particle  in  OB  is  tor,  where 
ft)  is  the  angular  velocity  and  r  the  distance  of  the  particle 
from  O. 


Hence 


and  Stnr^,  the  moment  of  inertia  of  the  rod  about  O,  is  Mz — ; 


4a^ 


.-.  ft,2=3_C 
2a 

which  gives  the  angular  velocity  of  OB. 

This  example  may  serve  also  to  show  the  independence  of 
the  motions  of  translation  and  rotation ;  for,  taking  the  expres- 
sion just  found, 

this  may  be  put  in  the  form 

Mga  =  }  M{a<of+ 1  lm{rco)^ 
where  ;'  is  measured  now  from  the  centre  of  inertia,  and  Sint^ 

=  M—  about  the  centre  of  inertia. 
3 

This  is  equivalent  to  saying  that  the  rod  in  dropping  from 
OA  to  OB  has  acquired  an  amount  of  energy  of  translation  of 
the  centre  of  gravity  (where  the  whole  mass  may  be  supposed 
to  be  collected)  equal  to  ^  M{aco)^,  and  also  an  amount  due  to 


S 


D'ALKMHKKT'S    I'RINCIPLE. 


4' 


rotation,  just  as  it  the  centre  of  gravity  were  fixctl  and  tiie  rod 

•J 
rotatinj^   about    it,  equal   to    \'^jfi{r(o)'^  or  .]•.)/'    -  (o^.     Or,  to 

put  it  in  another  way,  if  the  rod  were  freed  when  in  the  posi- 
tion O/i,  and  gravity  removed,  it  would  move  on  so  that  the 
centre  oi  gravity  would  have  a  velocity  {(to))  in  a  slrai.i,dU  line, 
and  wouKl  keep  rotatinj;  about  this  centre  of  jj;ravity  with  an 
angular  velocity  &> ;  and  the  two  energies  t.iken  together  would 
be  the  equivalent  of  the  work  tlone,  or  of  Jlj.nt. 

2.  A  uniform  stick  of  length  2  a  hangs  freely  by  one  end,  the 
other  end  being  close  to  the  ground.  An  angular  velocity  in  a 
vertical  plane  is  then  communicated  to  the  stick,  anil  when  it 
has  risen  through  an  angle  of  90',  the  end  by  which  it  was 
hanging  is  loosed.  What  must  the  initial  angular  velocity  be 
so  that  on  falling  to  the  ground  it  may  pitch  in  an  upright 
position  ? 

Figure  19  shows  three  positions  of  the  stick.  It  starts  with 
an  angular  velocity  &>,  communicated  to  it  in  some  way,  and 
reaches  its  second  position  with  an  angular  velocity  to',  such  that 


1 


(O" 


2  a 


(^^) 


a  relation  which  may  be  obtained  at  once  from  the  principle  of 
energy.     Then,  the  stick  being  freed,  the  centre  of  inertia  has  a 


;  from 
ion  of 
posed 
lue  to 


«/ 


cv 


I 
I 
A 

I 
I 
I 

4- 


■  •  t' 

-Si 


Fig.  19. 


ii 


:'!l   i 


42 


RIGID   DYNAMICS. 


motion  of  translation  upwards  represented  by  aco',  and  at  the 
same  time  the  stick  keeps  on  rotating  about  the  centre  of  inertia. 
Owing  to  the  action  of  gravity,  the  motion  of  translation  ceases, 
alters  in  cUrection,  and  finally  the  stick  drops  to  the  ground  in 
an  upright  position.  The  time  it  takes  the  centre  of  inertia  to 
move  from  its  second  position  to  its  final  position  when  the 
stick  pitches  upright  is  found  from  the  well-known  formula  for 
space  described  under  the  action  of  gravity,  which,  in  this  case, 
becomes 


a=  —aoi'- 1+\  (^fi. 


{b) 


The  condition  for  pitching  upright  is  evidently  to  be  found 
from  the  condition  that  the  rod  after  leaving  position  (2)  must 


7r 


rotate  through  (2//+  i)—  before  touching  the  ground,  and  there- 


fore 


w 


/  =  (2«+l) 


rr 


if) 


(a),  {b),  and  {c)  give  the  result 


„2=^ 


(-7^) 


where 


TT 


/=(2«+i)-. 
2 


3.  A  uniform  heavy  board  hangs  in  a  horizontal  position 
suspended  by  two  equal  parallel  strings  fastened  to  the  ends. 
If  given  a  twist  about  a  vortical  axis,  pro"e  that  it  will  rise 


a'-o) 


through  a  distance  — — ,  where  2  «  is  the  length  of  the  board,  and 


o)  the  vertical  twist. 


6^'- 


4.  A  cannon  rests  on  a  rough  horizontal  plane,  and  is  fired 
with  such  a  charge  that  the  relative  velocity  of  the  ball  and 
cannon  at  the  moment  when  the  ball  leaves  the  cannon  is  V. 
If  iM  be  the  mass  of  the  cannon,  in  that  of  the  ball,  and  fi  the 
coefficient  of  friction,  show  that  the  cannon  will  recoil  a  dis- 


tance  f-^ 

\M-{-in 


2^lg 


on  the  plane. 


I; 


D'ALEMBERT'S   PRINCIPLE. 


43 


(^) 


,  and 


fired 
and 

is  V. 
the 
I  dis- 


5.  A  fine  string  is  wound  around  a  heavy  grooved  circular 
plate,  and  the  free  end  being  fixed,  the  plate  is  allowed  to  fall 
freely.     Find  the  space  described  in  any  time. 

6.  A  coin  is  spun  about  an  axis  nearly  vertical  upon  an  ordi- 
nary table.  Form  the  equation  of  energy  at  any  time  as  the 
coin  descends  to  its  position  of  rest. 

7.  A  narrow,  smooth,  semicircular  tube  is  fixed  in  a  vertical 
plane,  the  vertex  being  at  the  highest  point ;  and  a  heavy  flexi- 
ble string,  passing  through  it,  hangs  at  rest.  If  the  string  be 
cut  at  one  of  the  ends  of  the  tube,  to  find  the  velocity  which 
the  longer  portion  will  have  attained  when  it  is  just  leaving  the 
tube. 

If  a  be  the  radius  of  the  tube,  /  the  length  of  the  longer  por- 
tion, then,  on  equating  the  kinetic  energy  at  the  time  the  string 
is  leaving  the  tube  to  the  work  done  by  gravity  up  to  that  time, 
it  will  be  found  that  the  required  velocity  is  given  by  the  relation 


'=^rt|2  7r-^(7r2-4)|. 


8.  Explain  why  the  grooving  in  a  rifle  barrel  diminishes  the 
force  of  recoil. 

9.  A  rough  wooden  top  in  the  form  of  a  cone  can  '•otate 
about  its  axis,  which  is  fixed  and  horizontal.  A  fine  string  is 
fastened  at  the  apex,  and  wound  around  it  until  the  top  is  com- 
pletely covered.  A  small  weight  attached  to  the  free  end  is 
allowed  to  fall  freely  under  the  action  of  gravity,  unwinding 
the  string  from  the  top  which  rotates  about  its  axis.  Find 
the  angular  velocity  of  the  top  when  the  string  is  completely 
unwound;  also,  the  equation  of  the  path  of  the  descending 
weight. 

10.  Two  equal  perfectly  rough  spheres  are  placed  in  unstable 
equilibrium,  one  on  top  of  the  other ;  the  lower  sphere  resting 
on   a   perfectly  smooth   horizontal   surface.      If   the   slightest 


'  1 

I 


;:i 


It 


44 


RIGID   DYNAMICS. 


disturbance  be  given  to  the  system,  show  that  the  spheres  will 
continue  to  touch  each  other  at  the  same  point,  and  form  the 
equation  of  energy  at  any  time. 

Figure  20  shows  the  solution  of  this  problem.  D'Alembert's 
principle  asserts  that  the  centre  of  inertia  must  descend  in  a 
straight  line,  since  the  only  external  force  is  gravity. 


ii 


^'''" 

~  ~''»v^ 

^ 

■V 

• 

\ 

/ 

N 

/ 

\ 

/ 

\ 

1 

\ 

1 
\ 
1 

\ 

\ 

\ 
1 

1 

1 

\ 

^^"^"^            /^^X. 

\ 

^r                     1                ^V 

\ 

/                       1                      ^V 

\ 

.  \ 

\ 

, 

y        \ 

N 

1 

/                                              V 

•V 

I 

X                                                                   I 

■^ 

-'"^'^                                                                   1 

■^ 

<-' 

"^  ""^^          ^^                               1 

Nv 

\^^^^                                       1 

/ 
/ 

3 

y<^\             1 

/ 

^ 

V                             \                             / 

1 

<e 

\       \      y 

1 

1 

~* 

1  Xs,^^^^^^^,^^ 

1 

1    ^"^  1  ""^ 

\ 

1        1 

\ 

f         1 

\ 

J 

/ 

\ 

/ 

■V 

^ 

/ 

X 

^ 

y 

'^.^ 

^ 

Fig.  20. 

Then,  considering  the  energy  of  translation  of  the  whole  mass 
of  the  two  spheres  collected  at  the  centre  of  inertia  along  with 
the  energy  of  rotation  of  the  system  about  the  centre  of  inertia, 
and  equating  the  sum  of  these  to  the  work  done  by  the  extci  iial 
force  of  gravity,  it  will  be  found  that  at  any  time  the  angular 
velocity  is  given  by 


rt2a)2(sin2  ^  + 1)  =  2  ^rt(  i  —  cos  6). 


;s  will 
n  the 


bert's 
in  a 


.,11 


CHAPTER   IV. 


'  ( 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES. 


33.  When  a  body  moves  in  such  a  way  that  two  points  in  it 
are  fixed,  this  is  equivalent  to  fixing  a  line  of  particles,  and  the 
motion  can  only  be  about  a  fixed  axis.  The  external  forces 
being  any  whatever,  these,  taken  along  with  the  pressures  on 
the  fixed  axis,  measured  in  the  proper  directions,  must  produce, 
according  to  D'Alembert's  principle,  equilibrium  with  the  re- 
versed effective  forces.  The  pressures  on  the  axis,  no  doubt, 
are  distributed  all  along  the  fixed  axis  and  vary  both  in  direction 
and  in  magnitude  from  point  to  point  and  cannot  generally  be 
determined  in  terms  of  known  quantities ;  but  it  is  usual  to 
suppose  that  they  may  be  regarded  as  equivalent  to  two  forces 
acting  at  two  definite  points  (which  may  be  chosen  anywhere 
along  the  axis),  and  in  particular  cases  to  a  single  pressure 
acting  at  a  symmetrical  centre. 


li 


i 


I'  '  f 


1''' 

"i 
1(1 


mass 
with 
;rtia, 
:j  nal 
ular 


34.    General  Equations  of  Motion. 

Let  the  body  be  an  extended  one  surrounding  the  point  O, 
as  in  Fig.  21;  let  ZOZ'  be  the  fixed  axis  about  which  rotation 
can  take  place,  and  the  plane  of  x?  the  plane  of  the  paper. 
The  axis  OY  (not  shown)  is  perpendicular  to  the  plane  of  the 
paper.  Let  the  pressures  on  the  axis  be  equivalent  to  P^  and 
P,^  acting  at  the  points  distant  .c^,  a^  from  the  origin  O,  and  let 
the  angles  which  these  pressures  make  with  the  axis  of  coordi- 
nates be  «!,  ^1,  7i,  ttj.  ^2.  72- 

4S 


i 


U 


46 


RIGID   DYNAMICS. 


Then  by  D'Alembert's  principle,  we  have 

%m  —^  =  SmX-i-  Pi  cos  a^  ■+■  P^  cos  «2. 

S w  ^  =  S;«  F+  /^i  cos  /Sj + Pa  cos  /S^, 

Sw  ~2  =  ^mZ^P^  cos  7i  +  /*2  cos  72, 
X,  F,  Z,  being  the  accelerations  on  the  unit  mass  m. 


Fig.  21. 

We  must  have,  also,  the  relations 


^^'YI?-  "  ^^)  "  ^  "^  ^1^1  ^^^  ^1  "*"  ^2-2  cos  /92, 
Sw/(  ^-^  -'^-:J;2  )  =  ^^±  ^l"l  ^O^  "1  ±  A'2^2  COS  ttg, 


where  L,  M,  N,  are  the  couples  produced  by  the  external  forces. 


^' 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES.       47 

It  will  be  seen  that  there  is  one  relation  independent  of  the 
pressures 


and  this  gives  at  once,  by  transformation  to  polar  coordinates, 

dt'^ 
and       .-.  gg^  moment  of  external  forces  about  the  fixed  axis 

df^  moment  of  inertia  about  the  fixed  axis 

which,  evidently,  on  integration  gives  the  angular  velocity  at 
any  time,  and  consequently  the  angle  described  in  any  o-iven 
time. 

35.    AiigHlar  Velocity  of  Any  Heavy  Body  about  a  Fixed  Hori- 
zontal  Axis. 

If  the  body  moving  about  a  fixed  horizontal  axis  be   acted 
upon  by  gravity  only,  the  angular  velocity  at  any  time  can  be 

^^       o 


Fig.  22. 


MM 
''1 


H'll 


It''' 


48 


RIGID   DYNAMICS. 


II' 


found  from  the  preceding  relation,  or  can  be  readily  deduced 
from  elementary  considerations  in  the  following  way. 

Let  the  heavy  body  be  movii  _,  about  an  axis  through  O 
(Fig.  22)  perpendicular  to  the  plane  of  the  paper.  It  may 
either  surround  O  or  be  rigidly  connected  to  it.  Let  the  plane 
of  the  paper  contain  the  centre  of  inertia  (A),  and  let  OA—h. 
Then,  if  at  any  time,  the  body  be  situated  as  represented,  the 
angle  between  the  vertical  and  OA  being  Q,  and  if  it  be  moving 
in  the  direction  of  B  increasing,  each  particle  of  the  body  is 
acted  upon  by  gravity  and  produces  a  couple  about  the  axis 
through  O. 

The  acceleration  of  each  particle   transversal  to  the  radius 


.    I  d 

vector  is , 

rdt 


r^ — ),  which  will  produce  a  couple 
dtj 


m 


S\^(y_ff\\ 


\  r  dt\     dtj  S 


|i 


about  the  axis  through  O.  D'Alembert's  principle  states  that 
when  we  sum  up  all  the  couples  produced  by  the  external  forces 
and  by  the  effective  forces  reversed,  we  must  obtain  equilibrium. 
Hence  we  must  have 


\  Wi^rj  sin  6^  +  m^gr^  sin  0^  -I- 


^MiJi 


id0 
dt 


Vr|=o, 


or 


'^{mgr  sin  6)-\-^mr^  •  -— =0. 


d^d_     Mgh  sin  Q 


dfi 


^inr^ 


which  is  the  same  relation  we  should  have  obtained  from  a  con- 
sideration of  the  general  equations  already  found.  Writing  this 
relation  in  the  form 


(//2  +  /^2) 


dfi' 


-gh  sin  ^, 


!1:  I ' 


MOTION    ABOUT   A   FIXED   AXIS.     FINITE   FORCES. 


49 


where  k  is  the  radius  of  gyration  about  a  parallel  axis  through 
the  centre  of  inertia,  and  integrating,  we  get 

(/^2  +  /.2)  A^y  .,^  2  ghicos  e  -  cos  a), 
or,  multiplying  by  the  mass  M, 


I  M{h^ + B)  ( 7'  J  =  Mgh{zos  0  -  cos  a), 

which  is  the  equation  of  energy ;  so  that  the  relation  obtained 
is  merely  another  way  of  expressing  the  equivalence  between 
work  done  by  gravitation  and  the  kinetic  energy  developed  dur- 
ing the  time  that  gravity  acted. 

36.    T/ie  Pendulum. 

If  we  suppose  any  body,  capable  of  motion  about  a  fixed 
horizontal  axis,  and  under  the  action  of  gravity,  to  be  slightly 
disturbed  from  its  position  of  equilibrium,  it  moves  to  and  fro, 
and  is  said  to  make  small  oscillations.  The  time  of  one  of  these 
excursions  can  be  found  from  the  expression  for  angular  accel- 
eration by  supposing  the  angle  6  so  small  that  sin  6  can  be 
represented  by  6.     Thus  we  have,  in  the  case  of  a  pendulum, 

which  represents  oscillatory  motion  and  periodic  values  of  Q, 
the  time  of  a  complete  oscillation  being 


Now  we  know  that  a  single  particle  suspended  by  a  weight- 
less string  of  length  /  will  make  small  oscillations  in  time 

2  7r\/- 
g 


lii 


i 


I 


!l 


I , 


'  (  1'  I 


50 


RIGID   DYNAMICS. 


I 


If,  then,  we  wish  to  find  the  length  of  a  simple  pendulum 
which  will  oscillate  in  the  same  time  as  an  extended  body,  we 
take 

k      ' 


/=' 


which  is  called  the  length  of  the  equivalent  simple  pendulum. 

Experimentally,  /  may  be  found  approximately  by  suspending 
near  the  body  a  simple  pendulum  made  of  a  small  heavy  body 
and  a  fine  string  whose  length  can  be  adjusted  until  the  times 
of  oscillation  of  the  two  are  the  same. 

37.    Centres  of  Suspension  and  of  Oscillation. 

Let  a  body  be  oscillating  under  the  action  of  gravity  about  an 
axis  through  5  perpendicular  to  the  plane  of  the  paper,  Fig.  23, 


Fig.  23. 

jp.  4_  J^i 

and  let  G  be  the  centre  of  inertia,  and   SO  =  l=-~^ — ,  the 

II 

length   of  the  equivalent  simple  pendulum.      5  is  called  the 
centre  of  suspension^  and  (7  the  centre  of  oscillation.     Now,  if 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES.        51 

the  body  be  inverted  so  that  it  can  oscillate  about  a  new  axis 
through  O,  then  the  new  length  /'  of  the  simple  equivalent  pen- 
dulum will  be  equal  to 


( 


li 


h 


=  1. 


Hence,  the  centres  of  suspension  and  of  oscillation  are  inter- 
changeable. 

38.  If  the  position  of  the  axis  of  oscillation  in  a  body  is 
changed,  the  time  of  oscillation  also  changes,  and  it  will  be 
found  that  this  time  is  a  maximum  when  the  axis  passes 
through  the  centre  of  inertia,  and  a  minimum  when  h  =  k, 
and  k  itself  is  a  minimum.  This  mav  be  seen  either  by  differ- 
entiating the  expression  for  /  or  by  throwing  it  into  the  form 


2k- 


{h-kf 
h 


\ ' 


:( 


Illustrative  Examples. 

I.    A  cube,  edge  horizontal  and  fixed,  makes  small  oscilla- 
tions about  this  edge. 

l=- a. 

3 


If  2^;  be  the  edge. 


2.  Find  the  time  of  a  small  oscillation  of  a  hemisphere  about 
a  horizontal  diameter  as  fixed  axis,  under  gravity. 

3.  A  wire,  bent  into  a  circle,  oscillates  under  gravity  (i) 
about  a  horizontal  tangent,  (2)  about  a  line  perpendicular  to 
this  tangent  at  the  point  of  contact.  Compare  the  times  of 
oscillation. 


52 


RIGID    DYNAMICS. 


4.  A  magnetic  needle  suspended  horizontally  by  a  fibre  with- 
out torsion  makes  small  oscillations  under  the  action  of  the 
earth's  magnetism.     Find  the  time  of  a  small  oscillation. 


>jb: 


M^ 


Fig.  24. 


If  //be  the  horizontal  intensity  of  the  earth's  magnetism,  m 
the  magnetic  strength  of  either  pole,  and  2  /  the  length  of  the 
needle,  then  it  is  evident,  from  Fig.  24,  that  when  the  needle 
makes  an  angle  0  with  the  magnetic  meridian,  we  have 


MJc^ 


df 


-Hvi  2  /sin  6?=  -HM'  sin  6, 


where  Mk"^  is  the  moment  of  inertia  about  the  axis  of  rotation, 
and  Al'  is  the  magnetic  moment. 
Hence,  for  small  oscillations, 


iPe    HM' 
dfi        I 


e=o, 


and  the  time  of  a  small  oscillation  is 


Tryj 


I 


HM' 


5.  A  circular  wire  carrying  a  current  and  freely  suspended, 
as  in  Ampere's  experiment,  places  itself  at  right  angles  to  the 
magnetic  meridian.  If  slightly  disturbed,  find  the  time  of  an 
oscillation. 

V.  a  solenoid  be  used,  find  also  the  time  of  a  small  oscillation. 

6.  Find  the  equation  of  motion  of  a  metronome  and  the  time 
of  a  small  oscillation. 


MOTION    ABOUT   A   FIXED   AXIS.     FINITK   FORCES. 


53 


7.  Find  the  least  axis  of  oscillation  for  a  sphere  and  an 
ellipsoid. 

8.  A  right  circular  cone  makes  small  oscillations  about  a 
diameter  of  its  base.  Find  the  time  of  one  of  these  oscillations, 
a  being  the  altitude,  and  b  the  radius  of  the  base.  Find,  also, 
the  least  axis  of  oscillation  when  a  — 2  b. 

9.  A  helix  of  wire,  with  the  ends  bent  inwards  and  ending; 
on  the  axis,  is  fastened  at  the  upper  end,  and  on  being  pulled 
slightly  by  the  lower  end  vertically  downwards,  and  then  freed, 
oscillates  under  gravity.     Find  the  time  of  an  oscillation. 

10.  A  uniform  beam  rests  with  one  end  on  a  smooth  hori- 
zontal table,  and.  has  the  other  end  attached  to  a  fixed  point  by 
means  of  a  string:  of  length  /.     Show  that  the  time  of  a  small 

7 
g 


oscillation  in  .  vertical  plane  will  be  2 


7r\/- 


II.  A  sphere  rests  on  a  rough  horizontal  plane  with  half  its 
weight  supported  by  an  extensible  string  attached  to  the  high- 
est point,  whose  extended  length  is  equal  to  the  diameter  of  the 
sphere.     Show  that  the  time  of  a  small  oscillation  of  the  sphere 

parallel  to  a  vertical  plane  is  2  7r'\/-^,  a  being  the  radius  of 
the  sphere.  ^^ 


12.  A  uniform  beam  of  length  2^  is  suspended  by  two 
equal  parallel  strings,  each  of  length  b,  fastened  at  the  ends, 
and  attached  to  fixed  points  in  the  same  horizontal  line.  Show 
that  if  given  a  slight  twist  about  a  vertical  central  axis  it  will 

^3^ 


make  small  oscillations  in  time  2  tt^ 


39.    Determination  of  g  by  the  Pcndnlmn. 

If  a  pendulum  of  any  form  be  allowed  to  make  small  oscilla- 
tions under  the  action  of  gravity,  wo  have  the  time  of  a  com- 


iff 


' 


ii    •' 


■     i 


54 


RIGID   DYNAMICS. 


plctc  oscillation  given  by  the  relation  /  =  2  7r'Y   .  where  /  is  the 

A''  /i^4-P 

length  of  the  equivalent  simple  pendulum  and  equal  to  — y — 

If,  now,  /  be  observed  by  means  of  a  clock,  and  //  and  /-  be 
found,  we  have  the  value  of  ^  given.  This  method  is  one  of 
the  most  accurate  known  for  finding  the  intensity  of  the  earth's 
attraction  at  different  points  on  its  surface.  Various  forms  have 
been  given  to  these  pendulums,  from  time  to  time,  in  order  to 
ensure  accuracy  of  measurement ;  and  the  most  important  of 
those  which  have  been  used  for  the  scientific  determination  of 
gravity  are  described  below. 

(a)  Berth's  Pendulum. 

Borda  (1792)  constructed  his  pendulum  so  as  to  realize  as 
nearly  as  possible  the  simple  pendulum.  It  was  made  of  a 
sphere  of  known  radius,  equal  to  a.  To  render  it  very  heavy  it 
was  composed  of  platinum  and  was  suspended  by  a  very  fine 
wire  about  twelve  feet  in  length.  The  knife  edge  which  carried 
the  wire  and  sphere  was  so  arranged  by  means  of  a  movable 
screw  as  to  oscillate  in  the  same  time  as  the  complete  pendulum. 

The  time  was  determined  by  the  method  of  coincidences^  and 
g  was  found  from  the  relation 


t=2lT 


where  /  is  the  length  from  the  knife  edge  to  the  centre  of  the 
sphere,  a  the  radius  of  the  sphere,  and  a  half  the  angle  through 
which  the  pendulum  swings  at  each  oscillation  to  or  fro. 

(b)    Katers  Pendulum. 

In  18 1 8,  Captain  Kater  determined  the  value  of  gravity  at 
London  by  applying  to  the  pendulum  the  principle  discovered 
by  Huyghens,  that  the  centres  of  suspension  and  oscillation  are 
reversible.     He  made  a  pendulum  of  a  bar  of  brass  about  an 


MOTION    AIJOUT    A    I'lXHD   AXIS.     FIMTK   KOUCKS. 


55 


inch  and  a  half  wide  and  an  LM;;hth  of  an  inch  in  thici<ncss. 
This  bar  was  pierced  in  two  places,  and  trianj^ular  knife  ed;;es 
of  hard  steel  were  inserted  so  that  the  distance  between  them 
was  nearly  39  inches.  A  lar^e  mass  in  the  form  of  a  cylinder 
was  placed  near  one  of  the  knife  edges,  being  slid  on  by  means 
of  a  rortangular  opening  cut  in  it.  A  smaller  mass  was  also 
attached  to  the  pen(hdum  in  such  a  way  as  to  admit  of  small 
motions  either  way.  The  i)endu]um  was  then  swung  about  the 
two  axes  and  adjustment  of  the  masses  made  until  the  times  of 
small  oscillations  were  the  same.  This  time  being  noted,  and 
the  distance  between  the  knife  edges  being  accurately  meas- 
ured, ,;,'•  was  readily  calculated.  A  small  difference  being  gen- 
erally found  in  the  two  times,  it  can  be  shown  that  the  length 
of  the  seconds  pendulum  will  be  found  from  the  expression 

(//i4-//a)(//i-//.,) 

where  //j,  h^  are  the  distances  of  the  centre  of  inertia  from  the 
two  knife  edges,  and  t^  t^  the  corresponding  times  of  oscillation. 

(c)    Rcpsold's  Pcndiiluin. 

It  was  noticed  in  experimenting  with  pendulums  made  like 
Kater's  that  the  vibration  is  differently  affected  by  the  sur- 
rounding air  according  as  the  large  mass  is  above  or  below. 
This  led  to  the  form  known  as  Rcpsold's,  in  which  the  two  ends 
are  exactly  similar  externally,  but  the  pendulum  (which  is  cylin- 
drical) is  hollow  at  one  end. 

The  centre  of  inertia  of  the  figure  is  equidistant  from  the 
knife  edges,  but  the  true  centre  of  inertia  of  the  whole  mass  is 
at  a  different  point. 

40.  Many  observers  have,  during  the  present  century,  con- 
ducted observations  at  different  points  on  the  earth's  surface 
in  order  to  determine  not  only  the  length  of  the  seconds  pendu- 
lum, but  also  the  excentricity  of  the  earth  considered  as  a 
spheroid. 


■I 


56 


RIGID   DYNAMICS. 


Hclmcrt  in  his  work  on  Geodesy  has  collated  the  results  of 
nearly  all  the  more  important  expeditions,  and  the  foUowin.ij^ 
table  gives  some  of  the  principal  stations  with  the  correspondini; 
lengths  of  the  seconds  pendulums  there,  and  the  name  of  the 
observer.     To  find  g  from  this  table  for  any  place,  the  relation 

log^'-=2log7r  +  log/ 


I ' 


K  '  i 


ll  I 


|i 


i!      ■     I 


Place. 


Latitude. 


Rawak .     . 

St.  Thomas 

(ialapagos 

Para      .     . 

Ascension . 

Sierra  Leone 

Trinidad    . 

Aden     .     . 

Madras 

St.  Helena 

Jamaica     . 

Calcutta     . 

Rio  Janeiro 

Valparaiso 

Montevideo 

Lipari  .     . 

Hoboken,  N. 

Tiflis     .     . 

Toulon 

Bordeaux  . 

Padua  .     . 

Paris     .     . 

Shanklin  Farm  (Isle  o 

Wight)  . 
Kevv  .  . 
Greenwich 
London 
Berlin  .  . 
Staten  Island 
Cape  Horn 
Leith  .  . 
Sitka  .  . 
Pulkowa  . 
Petersburg 
Unst      .     . 


o 
o 
I 

7 
8 

lO 
12 

'3 

15 

17 
22 
22 

2>Z 
34 
38 
40 

41 

43 

44 

45 
48 

50 
51 
51 
51 
52 
54 
55 
55 
57 
59 
59 
60 


I'S. 
24  N. 
32  N. 

27  S. 

55  S- 

29  N. 

38  N. 

46  N. 

4  N. 

56  S. 
56  N. 
iZ  N. 

55  S- 

2  S. 

54  S. 

28  N. 

44  N. 
41   N. 

7  N. 
50  N. 
24  N. 
50  N. 

37  N. 

28  N. 

28  N. 

31  N. 

30  N. 
46  S. 

51  s. 

58  N. 

3  N. 
46  N. 

56  N. 

45  N. 


/. 


99.0966 
99.1134 
99.1019 
99.0948 
99.1217 
99.1104 
99.1091 
99.1227 
99.1168 
99.1581 
99.1497 
99.1712 
99.1712 
99.2500 
99.2641 

99-3097 
99.3191 
99.3190 
99.3402 
99-3470 
99-3623 
99.3858 

99.4042 
99.4169 

99-4143 
99,4140 

99-4235 
99.4501 

99-4565 
99-4550 
99.4621 
99.4854 
99.4876 
99-4959 


Okskrver. 


Freycinet 
Sabine 
Hall 
Foster 


Sabine 


Basevi  and  Heaviside 
Basevi  and  Heaviside 


Sabine 

Basevi  and  Heaviside 


Liitke 
Foster 
Biot 


Duperrey 

Biot 

Biot 


Kater 


Foster 
Foster 


Lutkc 
Sawitsch 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES. 


57 


:s  of 

ding 

the 

ition 


may  be  used,  where  /  is  the  length  of  the  seconds  }iendulum 
in  centimetres.  See  also  Geodesy,  by  Colonel  A.  R.  Clarke, 
Chap.  XIV. 

The  places  are  arranged  geographically  in  order  of  their  lat;- 
t tides,  and  show  thereby  the  gradual  increase  in  the  length  of 
the  seconds  pendulum  as  we  go  from  the  equator  to  the  pole. 

Those  places,  in  the  preceding  table,  for  which  the  lengths 
of  the  seconds  pendulum  have  been  calculated  from  a  number 
of  observations  made  by  different  observers,  are  indicated  by  a 
dash. 


iide 
iide 


iide 


41.  During  the  past  few  years  several  observers  hav?  made 
observations  on  the  value  of  g  at  different  points  in  North 
America.  Professor  Mendenhall,  of  the  U.  S.  Coast  Survey, 
during  the  summer  of  1891,  visited  a  number  of  places  on  the 
Pacific  coast  between  San  Francisco  and  the  coast  of  Alaska, 
and  in  his  report  of  the  expedition  gives  a  table  of  the  values 
determined,  with  the  places  and  corresponding  latitudes.  He 
made  use  of  a  half-seconds  pendulum  enclosed  in  an  air-tight 
chamber  which  could  be  exhausted  with  an  air  pump.  A  spec- 
ial method  was  used  for  noting  the  coincidences  (see  U.  S.  Coast 
and  Geodetic  Sun>ey.     Report  for  1891,  Part  2). 

Defforges,  one  of  the  greatest  living  authorities  on  methods 
of  gravity  determination,  crossed  from  Washington  to  San 
Francisco  during  the  summer  of  1893  and  made  a  number  of 
observations  which  are  given  in  the  following  table.  The  value 
of  g  alone  is  given. 

Washington , 980.169 

Montreal 980.747 

Chicago 980.375 

Denver 980.983 

Salt  Lake  City 980.050 

Mt.  Hamilton 979.916 

San  Francisco 980.037 

These  are  all  reduced  to  sea  level. 


*1 


58 


RIGID    DYNAMICS. 


ll 


t'l  :- 


42.    Experimental  Determination  of  a  Moment  of  Inertia. 

In  many  cases  of  small  oscillations  under  gravity,  where  it 
is  difficult  to  calculate  the  moment  of  inertia  of  a  body  from  its 
elements,  the  time  of  oscillation  is  observed ;  and,  the  moment 
of  inertia  being  increased  by  the  addition  of  a  mass  of  definite 
figure,  the  time  of  oscillation  is  again  noted. 

The  required  moment  of  inertia  may  then  be  calculated. 

This  method  is  particularly  useful  in  the  case  of  magnetic 
oscillations  about  a  vertical  axis. 


lli, 


J' 

I?' 

11'  s 


'I 


43.    Pressure  on  the  Fixed  Axis, 
rical. 


Forces  and  Body  Symmet- 


If  a  body  be  moving  about  an  axis,  and  it  is  symmetrical  with 
respect  to  a  plane  passing  through  the  centre  of  inertia  and 
perpendicular  to  the  axis,  and  at  the  same  time  the  forces  acting 
on  the  body  are  also  symmetrical  with  respect  to  this  plane, 
then  we  may  suppose  that  the  pressures  on  the  axis  are  reduci- 
ble to  a  single  one  which  will  lie  in  the  plane  of  symmetry  and 
will  cut  the  axis  of  rotation.  To  determine,  in  such  case,  the 
direction  and  magnitude  of  the  resultant  pressure,  we  proceed 
in  the  following  way. 

Let  the  body,  Fig.  25,  surround  the  point  O  and  let  it  be 
symmetrical  with  respect  to  the  plane  of  the  paper  which  con- 
tains C,  the  centre  of  inertia :  the  axis  of  rotation  being  perpen- 
dicular to  the  plane  of  the  paper,  and  passing  through  O.  Let 
the  forces  acting  on  the  body  also  be  symmetrical  with  reference 
to  this  plane.  And  let  the  body,  moving  about  the  axis  through 
O,  be  situated  at  any  time  t  as  represented,  Q  being  the  angle 
which  the  line  OC  fixed  in  the  body  and  moving  with  it  makes 
with  the  line  OA  fixed  in  space.  Then  the  resultant  pressure 
on  the  axis  will  be  in  the  plane  of  the  paper,  and  its  direction 
will  pass  through  O.  Let  its  components  measured  along  two 
rectangular  axes  OX,  0Y\\\  the  body,  be  /'and  Q.    Let  CO  =  Ji. 

Then,  X,    V,  being  the  accelerations  on  unit  mass  in  the 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES. 


59 


I  -! 


directions  OX,  OY,  we  have,  by  D'Alembert's   principle,  the 
relations 


Fig.  25. 


Fig.  26. 


Now,  if  Q)  be  the  angular  velocity,  any  particle  such  as  ;;/ 
will  be  acted  on  by  the  forces  wwV,  mwr,  as  is  indicated  in  the 
figure;  and  these  forces  resolved  along  OX,  OY,  as  shown  in 
Fig.  26,  would  give 

in  -7-  =  —  mw^x  —  niwy, 

'ib'  2    ,       • 

tn  —f-  —  —  viwy  +  inoix. 

dr 

The  values  of  ~,  ^---^  may  also  be  obtained  by  direct  differ- 
entiation from  .r=rcos^,  j  =  rsin^.     Thus, 

^'V  •     .d9  dy  ^de 

~-^-  —  r  cos  6   .  =,i-ft). 


- .-  =  —  ^'  Sin  6  —  z=  -yco, 

dt  dt         -^   '         dt 


dt 


\ 


I 


■  ii 


H  l! 


M      \ 


60 


RIGID    DYNAMICS. 


Hence,  our  relations  for  determining  the  pressi'"es  become 

P  +  '^mX-\-  l.m{oP'x  +  wj')  =  o, 

Q  +  2;«  F+  2;;/  (tu'-^j  —  iox)  =  O. 

. '.    /-'  =  —  2;;/ A'—  2 w  (tu^.r  +  &>J')> 

Q=  —"^jii  Y—  2 w (w^j  —  w.i'). 

But,  by  definition  of  the  centre  of  inertia, 

2wa)^.r=ft)22  'nx=MJioiP',    '^vmy  =  io^niy  =  0, 

^mvP'y  =  (jiP'^my  =  o,    '^inwx  —  io^inx=  Mhto. 

.:   P=-^mX-M]m\ 

Q=-'LmY+Mhio, 

which  equations  determine  the  pressures  P,  Q,  and  therefore 
the  direction  and  magnitude  of  the  resultant  pressure  when 
we  know  q>,  which  is  found  from  the  relation  already  given, 

.      d^0       N 
ft)  =  — ~  = 


where  A^  is  the  moment  of  the  external  forces  about  the  rota- 
tion axis,  and  '2nir^  is  the  moment  of  inertia  about  the  same 
axis.  This,  on  integration,  gives  <y,  and  on  substituting  its 
value  in  the  preceding  expression,  P  and  Q  are  found. 


I 


) 


44.   Heavy  Symmetrical  Body.     Pressure  on  the  Axis. 

In  the  particular  case  of  a  heavy  body  which  is  symmetrical 
about  a  plane  through  its  centre  of  inertia  perpendicular  to  the 
rotation  axis,  which  is  horizontal,  the  external  forces  are  only 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES.        6 1 

those  of  gravity,  and  we  have,  Fig.  27,  the  pressures  given  by 
the  relations 

P  =  -  Mg  cos  e  -  MJm\ 

Q  =  AIgs\r\d  +  Mhw, 

and  if  we  suppose  P  estimated  in  the  opposite  direction,  the 
complete  solution  of  the  motion  is  obtained  from 


)\ 


Fig.  27. 


d(o  _  dW  _  _  gh  si n  ^ 
~dt~~dfi~     Ifi  +  W 
P  =  Mg  cos  e  +  M/m"^, 
Q  =  Mg&\ne  +  M/iw, 


9  being  measured  always  upwards  from  the  vertical  and  k  being 
the  radius  of  gyration  about  the  centre  of  inertia. 


m  i 


I'll  I 


i  ■  'II 


I*!; 


t  i 


t  » 


I 
I 

i  ' 
|S     I   j 


i 


62 


RIGID   DYNAMICS. 


Illustrative  Examples. 


I.  A  rod,  movable  about  one  end,  falls  in  a  vertical  plane, 
starting  from  a  horizontal  position.  Find  the  pressure  on  the 
end  in  any  position. 

Figure  28  shows  the  motion  ;  when  the  rod  rtiakes  an  angle 
Q  with  the  vertical  line  OA,  we  have 

(l(o _d^d _     ga  sin  6 


and 


dt 


=  --^-^sin^. 


day 
2  (O  -  = 
dt 


3.^ 


4  a 


sin  6  ,- 
2  a  dt 


.'.     I    2Wa)=—  I     — -SI 

Jo  »^!r  2  a 


sin  Ode. 


Fig.  28. 


ft)' 


: cos  ^, 

2  a 


P = Mg  cos  ^  +  i^A70)2  =  \  JSIg  cos  ^, 
(2  =  Mg  sin  ^  +  Maii  =  \  Mg  sin  6. 


\' 


MOTION   ABOUT    A   FIXED   AXIS.     FINITE   FORCES.        6^ 

When  the  rod  i.s  in  the  lowest  position,  6'  =  o,  and  r^'>  Af<r 
<2=o.  •-    "' 

2.    Rod,  movable  about  one  end,  falling  from  the  position  of 
unstable  equilibrium. 

As  in  the  preceding  problem,  we  have  (Fig.  29) 

(It         4  a 

«'  =  |f(i+cos^), 

P  =  Mg  cos  e  +  J/.?a)2  =  1  J/^(3  +  5  cos  6), 
Q  =  Mg  sin  e  +  l^^aw  =  \  Mg  sin  6. 


and 


Fig.  29. 


In  the  lowest  position,  e  =  o,  Q  =  o,  P  =  ^Mg,  which  shows 
that  if  the  rod  can  just  make  complete  revolutions,  the  pressure 


i"j 


i  ■  a 


64 


RIGID   DYNAMICS. 


on  the  axis  in  the  lowest  position  is  in  the  direction  of  the  rod, 
and  equal  to  four  times  its  weight. 

Maximmn  and  Minininin  Values  of  P  and  Q. 

P  is  a  maximum  when  d=n  or  t,  and  its  values  then  are 
^Mg  and   —Mg\   it  is  a  minimum  when  cos^=  — |.     (^  is  a 

maximum  when  6  —  —.  and  it--  vaUie  then  is  \j\Ig\  it  is  a  mini- 
mum  when  d  —  o  '■■r  tt,    '  'I        value  then  is  o. 


Resultant  Pressure  at  ^x,,y  Ti.nc 

This  maybe  found  by  taking  R'^=P'^-\-Q^,  and  substituting 
the  general  values  of  P  and  Q  in  terms  of  6.  The  maximum 
and  minimum  values  of  the  total  pressure  may  be  obtained  by 
differentiating  in  the  usual  way.  The  angle  which  the  result- 
ant pressure  makes  with  the  rod  will  be  determined  from  the 
■  _<2_       sin^ 


relation  tan  o/r: 


P    6+IOCOS0 


3.    Cube,  edge  horizontal,  performing  complete  revolutions 
under  gravity. 


O    ct 


Fig.  30. 


Figs.  30  and  31  show  the  motion.  Since  the  body  and  forces 
are  symmetrical  about  the  central  plane  perpendicular  to  the 
axis  of  rotation,  the  pressures  on  the  axis,  as  the  cube  swings 


fc 


ar 
th 


: 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE    FORCES.       65 

around,  are  reducible  to  a  single  pressure  lying  in  this  central 
plane  and  cutting  the  axis.     Taking,  then,  the  auxiliary  figure, 
we  need  only  consider  the  motion  of  OC,  which  in  any  position 
makes  an  angle  0  with  the  vertical  line  OA. 
The  angular  velocity  at  any  instant  is  given  by 

dan _d'^d _         ^  q- 

the  edge  of  the  cube  being  of  length  2  a. 

Supposing   the   cube   to   start   initially    with    OC  vertically 
upwards,  and  to  swing  completely  around, 


•.    I  2  codoj  =-  C  -^^  sin  ed0, 


«^=-~-(i+cos(9), 


2V2 


a 


and 


P=A/^cos  e  +  Ma^2(^^-i  +c^, 

\~-\2a  J 


Q  =  M^ sin  e+A/aV2( ^ 


f-  sin  d\ 
2a         J 


4V2 
•••  ^=-/(3  +  Scos^), 

Q  =  ~~smd. 
4 

The  maximum  and  minimum  values  of  P  and  Q  can  easily  be 
found,  as  in  the  previous  case  of  the  rod. 
To  find  the  total  presstu'c,  we  have 

^^=/^He^=(fJ(3  +  5cos.)^+(^Jsin^,, 

and  the  maximum  and  minimum  values  of  R  can  be  found  by 
the  process  of  differentiation  with  respect  to   6.     It  will  be 

F 


w 


i 


I 


66 


RIGID   DYNAMICS. 


found  that  R  is  maximum  when  ^  =  o  or  wtt,  and  its  values  then 
are  4^14'  ^^^^^  —-^^g- 

A'  is  a  minimum  \\  hen  cos  6=  ~  ^!]»  ^^^^  its  value  is  -  '-^\  .   • 

4.  A  hemisphere  revolves  about  an  axis  which  coincides  with 
a  diameter  of  its  base,  and  which  is  inclined  at  an  angle  «  to 
the  vertical.  If  it  swings  completely  around,  the  total  pressure 
on  its  axis,  when  in  the  lowest  position,  is 

IV  i 

--  K 1 09  sin  «)2  4-  (64  cos  (if  I  \ 
64 

5.  A  right  circular  cone  whose  height  is  equal  to  the  radius 
of  its  base  swings  about  a  horizontal  axis  through  its  vertex. 
If  the  axis  of  the  cone  starts  from  a  horizontal  position,  find  the 
angular  velocity  and  the  pressure  on  the  rotation  axis  when  in  the 
lowest  position. 

6.  A  uniform  heavy  rod  oscillates  about  one  end  in  a  vertical 
plane,  under  gravity,  coming  to  rest  in  a  horizontal  position. 
If  i/r  be  the  angle  between  the  rod  and  the  line  of  the  resultant 
pressure,  and  ^the  angle  of  inclination  of  the  rod  to  the  horizon 
at  the  same  time,  then  tan  ^fr  tan  ^=  jV 

7.  A  homogeneous  solid  spheroid,  the  equation  of  whose 
bounding  surface  is 


is  suspended  from  an  axis  passing  through  one  of  the  foci. 
Prove  that  the  centre  of  oscillation  lies  on  the  surface 


m 


\  a\x^  +  b\x^  -{-f  +  .C-2)  1 2  =  2  5  «2(^2  _  ^2^  (1-2  j^f  4.  .2)^_ 

8.    A  uniform  wire  is  bent  into  the  form  of  an  isosceles  tri- 
angle, and  revolves  about  an  axis  through  its  vertex  perpendicu- 


MOTION   ABOUT   A   FIXED   AXIS.     FINITE   FORCES. 


^7 


lar  to  its  plane.     I'rovc  that  the  centre  of  oscillation  will  be  at 
the  least  possible  distance  when  the  triangle  is  right  angled. 

9.  A  uniform  heavy  rod  revolves  uniformly  about  one  end  in 
such  a  manner  as  to  describe  a  cone  of  revolution.  Find  the 
pressure  on  the  fi.xed  point,  and  show  that  if  0,  yjr  be  the  angles 
which  the  vertical  makes  with  the  rod  and  the  resultant  press- 
ure, 4  tan  ■«/r  =  3  tan  0. 

10.  A  rough  uniform  board  is  placed  on  a  horizontal  table 
with  two-thirds  of  its  length  projecting  over  the  table,  the 
board  being  initially  in  contact  with  the  table,  and  perpendicu- 
lar to  the  edge.     Show  that  it  will  begin  to  slide  off  when  it 

has  turned  through  an  angle  tan"^  -,  (jl  being  the  coefificient  of 
friction. 

45.    General  Case. 

If  the  forces  and  body  arc  not  symmetrical,  then  we  take  the 
general  equations  already  found  ;  and  supposing  the  pressures 
to  be  equivalent  to  two  at  two  points  on  the  axis  whose  com- 
ponents are  /',  (2,  A' ;  P\  Q' ,  R'  ;  we  get,  for  the  determination 
of  these  pressures,  the  relations 


foci. 


2.m 


2w 


ir-z 


_  V 


d{' 


-IwZ+R  +  R', 


dfi    -^  dt^J  dt 


I  , 


68 


Kir.ID    DYNAMICS. 


This  l;ist  relation  gives  at  once  the  value  of  w  and,  by  integra- 
tion, of  0).  /-,  J/,  N  arc  the  couples  i^roduced  by  the  external 
forces  ;  and  C^,  C^  C{,  C,l,  the  couples  produced  by  the  pres- 
sures, which  can  be  expressed  in  terms  of  these  pressures,  and 
the  distances  from  the  origin  at  which  they  arc  supposed  to  act. 

The  process  of  solving  any  particular  problem  will  be  to 

(i)    JMnd  (o  and  o). 

(2)  Express  the  quantities    -  '- ,  etc.,  in  terms  of  rl),  ro,  and 

dr 
known  expressions. 

(3)  Thence  find  the  pressures. 

The  effective  forces  can  be  expressed  in  terms  of  the  radial 
and  transversal  forces  either  by  resolution  or  by  direct  differ- 
entiation, and  it  will  be  found  that 

d\\- 

db'  0    , 

—    =  — 'W-j'  +  w.r. 


l!    ' 


'    -I 


Thus,  the  previous  relations  will  become 

2w^V+  P  +  /"  =  2///  ( -  0)2  V  -  ioy!)  =  -  (oW.v  -  (oMy, 
2;«  Y+Q  +  Q'  =  S;;/(  - afiy  -f  wx)  =  - a^My  +  (oMx, 
2wZ  +  A'  +  /v"=o, 
where  x,  y  are  the  coordinates  of  the  centre  of  inertia. 

Also,  since      —"=0,    we  have 
dt^ 

Z  +  r,  +  C  =  2;;/  y  — ^  —  z  -f-  ]  =  ay^^mya  —  wSw/a'^, 
\    dr        drj 


Wi 


M^C(^Cl  =  ^vt[rJ^^-x 


d^y 


d^d 
dfi 

d'ir\ 


=  —  (o^'Linx:::  —  od^inyz, 


^<PS 


A'=2«(.v^-^'^-J  =  2,«.    ^^ 


MOTION    AllOUT   A    I'lXKI)   AXIS.     FINITK    FORCKS. 


69 


46.  It  will  be  seen  that  the  last  expressions  are  much  simpli- 
fied if  wj  make  a  proper  choice  of  axes.  The  first  thin^-  to  be 
done,  then,  is  to  choose  the  ori;;in  on  the  rotation  axis  so  that 
it  is  a  principal  axis  at  that  point.     Then  '^p/xc^o,  ^tfn'.::  =  o. 

Thus,  for  example,  if  we  su|)pf)se  a  triant^le  to  be  rotating 
under  gravity  about  one  side  which  is  horizontal,  the  equa- 
tions of  motion  will  be  much  simjilified  if  we  choose  as  (^rigin 
that  point  in  the  side  at  which  it  is  a  principal  axis  ;  see  Kx.  3, 
p.  28.  Then,  supposing  the  pressures  to  be  equivalent  to  two 
acting  at  the  ends  of  the  side,  'Vie  solution  is  very  simple,  as 
the  angular  velocity  at  any  time  is  found  from  the  relation 


I!' 


^  sin  e 

3 

6 


2  i,--  sin  0 


111! 


where/  is  the  perpendicular  on  the  rotation  axis  from  the  oppo- 
site angle ;  and  the  pressures  can  then  be  immediately  written 
down  in  terms  of  w,  to  and  the  coordinate  x  of  the  centre  of 
inertia,  y  being  o,  since  the  body  is  a  lamina. 


.1  :'i 


CHAPTER   V. 


MOTION   ABOUT   A   FIXED   AXIS.     IMPULSIVE   FORCES. 


an  I 

'  I 


47.    General  Case. 

An  impulse  being  defined,  as  already  explained,  to  be  a  force 
which  produces  a  sudden  change  of  velocity,  and  which  only 
acts  for  an  indefinitely  short  time,  we  can  obtain  the  general 
impulsive  equations  of  motion  of  any  body  capable  of  motion 
about  a  fixed  axis  by  considering  the  relations  found  in  Art.  45. 
In  those  relations,  by  the  substitution  of  changes  of  velocity 
for  accelerations,  we  get 


i     1; 


2A'+/^  +  /^'  =  2wK'''-''^^' 


Wdt 


dt 


2F+G  +  (?'  =  Sw 


-v,J,^^''''Y 


\  \d~t 


dt)  S ' 


im  ■  <i 


^Z  +  R  +  R'^o, 

where  X,  Y,  Z  are  the  impulsive  actions  on  individual  parti- 
cles due  to  external  impulsive  forces  ;  /',  Q,  R,  P' ,  0',  R'  are 
impulsive  pressures  on  the  fixed  axis  ;  and  where  the  velocity 

f '^4  !  of  any  particle  before  the  impulsive  action  takes  place,  is 


\d}j 

changed  suddenly  to 


70 


5  ( 


MOTION    ABOUT   A   FIXED   AXIS.     IMPULSIVE   FORCES.    71 

And,  since  -j-  =  o,  and    the   angular   velocity  w  is    suddenly 
at 

changed  to  «',  we  have  for  the  impulsive  couples, 

Now  we  have  ~   at  any  time  equal  to   —wy,  and    --^  =  &),r; 
at  (it 

and,  substituting  these  values  in  the  preceding  equations,  we 

have  the  complete  solution  of  the  problem  given  by 

2 A>  P  -f  .P'  =  -  2;;/  («'  -  ft))j/  =  -  (w'  -  a))iW>, 

2  F+  S  +  g'  =  2;/^ («'  - ft)),r=  (o)'  - (o)M~x, 

'1Z  +  R  +  R'=0, 

L  +  C-^  +  C<2^=  —  S  w.c(&)'  —  (o)x=  —  (o)'  —  ft))  ^nixs, 

M+  C^  +  C2'  =  2;«^(ft)'  -(w)j=  (ft)'  -ft))Swj.?, 

7V=  (ft)'  —  ft))  •  "^Mf^. 

48.  If  the  body  starts  from  rest,  then  co  =  o,  and  the  sudden 
angular  velocity  generated  by  an  impulse  which  tends  to  turn  a 
body  about  a  fixed  rotation  axis  is  obtained  from  the  relation 


'1' 


n 


'f  •'] 


ft) 


)lacc,  is 


where  N  is  the  moment  of  the  impulse  about  the  axis,  and 
^m)'^  is  the  moment  of  inertia.  As  before,  the  problem  is  sim- 
plified by  choosing  the  origin  at  a  point  where  the  rotation  axis 
is  a  principal  axis. 


I    ! 


'mi  i 


72 


RIGID   DYNAMICS. 


49.    Centre  of  Percussion. 

In  the  general  equations  just  found,  let  us  suppose  that  the 
impulsive  actions  ?re  those  caused  by  a  blow  Q  represented  by 
components  X,  F,  Z\  and  that  the  blow  is  struck  at  some  point 
on  the  surface  of  a  body,  capable  of  motion  about  a  fixed  axis, 
which  either  passes  through  it  or  to  which  it  is  rigidly  con- 
nected. What  is  the  condition  that  there  shall  be  no  impulsive 
pressure  on  the  axis  ?  Or,  in  other  words,  is  it  possible  to 
strike  the  body  at  a  certain  point  in  such  a  way  as  to  produce 


Fig.  32. 


no  strain  upon  the  axis  about  which  it  is  free  to  rotate }  Let 
the  body  (Fig.  32)  surround  0\  let  ZZ^  be  the  axis  of  rota- 
tion, and  let  the  plane  of  zx,  which  is  the  plane  of  the  paper, 
contain  G,  the  centre  of  inertia  of  the  body.  Suppose  that  the 
blow  Q  is  applied  at  the  point  whose  coordinates  are  ^,  ?;,  ^  (the 
coordinate  r\  not  being  shown,  being  drawn  upwards  perpen- 
dicular to  the  plane  of  the  paper).     If  there  be  no  resulting 


b 

P' 

C( 


Let 
rota- 
aper, 
It  the 
^(the 
rpen- 
ilting 


MOTION   ABOUT   A   FIXED   AXIS.     IMPULSIVE   FORCES. 


73 


pressure    when    the    body   is    struck,    the    general    relations 
become  : 

X=o, 

Y={(o'-co)Mv, 

Z=o, 

L  =  qZ—  ^  V=  —  {(!)'  —  oi)^mx3y 

M=  ^X-  ^Z=  -  {co'-o})'2mj'::, 

N=^V-r]X=  {(o'  -(o)^m>'^={oi'  -(o)Mk\ 

where  k  is  the  radius  of  gyration  about  the  axi.s. 

From  these  it  will  be  seen  that,  since  X=o,  Z=o,  we  have 
also  2wjxr=o.     And  also, 

^  Y=  («o'  —  Q))^mxc, 

Y=:{(0'-0))MX. 

-,  _  2  ;;/xc  _  2  jhx::: 
Mx        "Lmx 

And  f  is  given  by  the  last  relation, 

t^jco'-  o>)MP  ^{(o'-  co)MP  ^  P 
V  {a)'-(o)Mx      X 

The  above  conditions  holding,  and  there  being  no  pressure  on 
the  axis,  the  line  of  the  blow  is  called  a  Line  of  Percussion,  and 
any  point  in  this  line  is  termed  a  Centre  of  Percussion. 

50.    By  an  inspection  of  the  foregoing  relations,  we  have, 

I.  A'=o,  Z—0\  and  therefore  one  condition,  that  there  may 
be  no  strain  upon  the  axis,  is  that  the  line  of  the  blow  must  be 
perpendicular  to  the  plane  containing  the  rotation  axis  and  the 
centre  of  inertia. 


;;i:i. 


n  I 


74  RIGID   DYNAMICS. 

2.  2;;y^  =  o,  and  S;;/;r^  =  ^'  2;«;r. 

Now,  since  O  may  be  chosen  anywhere  on  the  axis,  let  it  be 
so  chosen  that  ^=o.  Then  for  that  origin  so  chosen  ^niyz 
would  be  zero,  and  ^mxs  also  zero. 

Therefore,  an  essential  condition,  to  be  first  satisfied  for  a 
line  of  percussion,  is  that  the  axis  of  rotation  must  be  a  prin- 
cipal axis  at  some  point  of  its  length. 

3.  f =-->  which  shows  that  when  a  centre  of  percussion  does 

exist,  its  distance  from  the  axis  is  the  same  as  that  of  the  centre 
of  oscillation. 

If  ^=0  and  3=0,  then  the  line  of  percussion  passes  through 
the  centre  of  oscillation,  which  may  be  stated  in  the  following 
way : 

//  tJic  fixed  axis  be  parallel  to  a  principal  axis  at  the  centre  of 
inertia,  the  line  of  action  of  the  bloiv  will  pass  through  the  centre 
of  oscillation. 

Illustrative  Examples. 

I.  A  uniform  rod,  fixed  at  one  end  and  capable  of  motion  in 
a  vertical  plane,  is  hanging  freely  under  .i.  action  of  gravity, 
and  being  struck  perpendicular  to  it^  length,  rises  into  the 
position  of  unstable  equilibrium.  Find  the  magnitude  of  the 
blow  that  there  may  be  no  strain  nt  the  fixed  point. 

In  order  that  there  may  be  no  strain  on  the  axis,  it  must  be 
struck  at  the  centre  of  percussion,  which  point  will  be  at  a 

distance  4.-  from  the  fixed  end,  if  the  length  of  the  rod  be  2  a. 

3 
Then,  if  w  be  the  angular  velocity  produced  by  the  impulse,  we 

ha^c  from  the  equation  of  moments. 


3 


\,vir 


(O. 


B=Maw. 


MOTION   ABOUT   A   FIXED   AXIS.     IMPULSIVE   FORCES. 


75 


Also, 


—  =  —  --'^  sin  6 


is  the  equation  of  motion  of  the  rod  as  it  rises  upwards,  being 
acted  upon  by  gravity,  and  starting  with  an  angular  velocity  «. 


«/(o  2il*^^ 


0) 


2_ 


3^, 

a 


.'.   B  =  LI  aw  =  J/V  3  ga. 

From  this  it  may  be  seen  that  generally  whe:^  a  body  is 
struck  at  the  centre  of  percussion,  the  value  of  the  impulse  is 
measured  by  the  product  of  the  mass  and  the  velocity  of  the 
centre  of  inertia. 

2.  A  cu'cular  plate  free  to  move  about  a  horizontal  tangent 
is  stuck  at  its  centre  of  percussion  and  rises  into  a  horizontal 
position.     Find  the  blow. 


As  before, 
and 


B^Maoi,     a  being  the  radius, 

d(iy  AiT   •     a 

— ^-^  sm  6     gives  o). 


dt 


5^ 


^   5 

3.  A  sector  of  a  circle,  whose  radius  is  a  and  ingle  «,  is 
capable  of  turning  about  an  axis  in  its  plane  which  is  perpen- 
dicular to  one  of  its  bounding  radii.  Find  the  coordinates  of 
the  centre  of  percussion. 

f'^'g-  33  shows  the  position  of  the  centre  of  percussion  C, 
whose  coordinates  are 


^mx 


2.1HX- 


2,7/ /X 


>     »>-Vll  u 


v.\^'^,",. 


|!     il 


76 


RIGID   DYNAMICS. 


On  transforming  to  polar  coordinates  it  will  be  found  that 

^=  \a  sin  «, 


f=^^b 


(  — 1-  cos  «  ). 

Vsin  / 


Fig.  33. 


4.    To   find  the  centre  of   percussion   of   a  triangular  plate 
capable  of  rotation  about  a  side. 


Fig.  34. 


MOTION   ABOUT   A   FIXED   AXIS.     IMPULSIVE   FORCES. 


77 


Fig.  34  shows  the  position  of  the  centre  of  percussion.  AB 
is  the  rotation  axis,  PD  perpendicular  to  AB,  E  the  middle 
point  of  AB,  F  the  middle  point  of  DE.  Then  AB  is  a  prin- 
cipal axis  at  the  point  F,  and  G  being  the  centre  of  inertia  of 
the  plate,  and  PD—p, 

C  is  the  centre  of  oscillation, 

C  is  the  centre  of  percussion, 


and 


h       p       2 

3 


When  the  triangle  is  isosceles,  C  and  C  coincide. 

5.  ABCD  is  a  quadrilateral  (Fig.  35),  AB  being  parallel  to 
CD.  Show  that,  if  AB^—iCD^,  the  point  /*  is  a  centre  of  per- 
cussion for  the  rotation  axis  AB.     (Wolstenholme.) 


ill 


Fig.  35. 


6.  A  uniform  beam  capable  of  motion  about  one  end  is  in 
equilibrium.  Find  at  what  point  a  blow  must  be  applied  per- 
pendicular to  the  beam  in  order  that  the  impulsive  action  on  the 
fixed  end  may  be  one-third  of  the  blow. 


7« 


RIGID   DYNAMICS. 


51.    Initial  Motions.     Changes  of  Constraint. 

If  a  body,  moving  about  a  fixed  axis  with  known  angular 
velocity,  is  suddenly  freed  from  its  constraint  and  a  new  axis 
fixed  in  it,  or  if  a  body  at  rest  is  disturbed  so  that  there  is  a 
sudden  impulsive  change  of  pressure,  we  can  determine  the 
new  angular  velocities  and  changes  of  pressure  by  reference  to 
the  impulsive  equations  of  motion  already  found.  Sometimes, 
however,  solutions  which  are  more  instructive  may  be  obtained 
by  cc  idering  elementary  principles  ;  and  the  following  exam- 
ples are  given  to  illustrate  the  methods  to  be  employed  in 
various  cases. 


II    !!> 


Illustrative  Examples. 

I.  A  uniform  board  is  placed  on  two  props;  if  one  be  sud- 
denly removed,  find  the  sudden  change  in  pressure  at  the  other. 

Fig.  36  illustrates  the  problem.  The  board  is  of  length  2  a, 
and  rests  on  the  props  A  and  B,  which  are  fixed  in  position  in 


['I 


a, 


^ 


Cty 


M^ 


M 


B 


Fig.  36. 


JR' 


A 


My 


W  i    I 


the  first  figure,  so  that  R  =  \Mg.  If  B  be  now  removed,  the 
board  begins  to  turn  about  the  upper  end  of  A  under  the  action 
of  gravity,  and  to  each  element  of  the  board  an  acceleration  wr 
is  given  suddenly  ;  so  that  if  we  communicated  to  each  element 
7n  an  acceleration  ar  in  the  opposite  direction  (upwards),  we 


MOTION   ABOUT   A    FIXED   AXIS.     IMPULSIVE    FORCES.      79 

would  have,  by  the  p,pplication  of  D'Alembert's  principle,  A", 
2L(;;/aj;'),  and  Mg  in  equilibrium  with  one  another,  as  indicated 
in  the  second  figure. 

Also,  taking  moments  about  O,  just  when  the  prop  is  removed, 
we  have 

'^{mcor)  •  r=Mg-  a. 


(0  = 


3f 
4  a 


.:  R'  =  Mg  -  Maco  =  \I\fg. 

2.  The  extremities  of  a  heavy  rod  are  attached  by  cords  of 
equal  length  to  a  horizontal  beam,  the  cords  making  an  angle  of 
30°  with  the  beam.  If  one  of  the  cords  be  cut,  show  that  the 
initial  tension  of  the  other  is  two-sevenths  of  the  weight  of  the 
rod. 

3.  A  uniform  rod  is  suspended  in  a  horizontal  position  by 
means  of  two  strings  which  are  attached  to  the  ends  of  the  rod. 
If  one  of  these  strings  be  suddenly  cut,  find  the  sudden  change 
in  tension  of  the  other  string. 

4.  Two  strings  of  equal  length  have  each  an  extremity  tied 

to  a  weight  C,  and  their  other  extremities  tied  to  two  points 

A,  B  in  the  same  horizontal  line.     If  one  be  cut,  the  tension  of 

ACB 
the  other  is  instantaneously  altered  in  the  ratio   i  :2  cos^ . 


5.  A  particle  is  suspended  by  three  equal  strings  of  length  a 
from  three  points  forming  an  equilateral  triangle  of  side  2  ^  in  a 
horizontal  plane.     If  one  string  be  cut,  the  tension  of  each  of 

the  others  is  instantaneously  changed  in  the  ratio  ^- —^ — 

^  ""  2{d'-0'') 


80 


RIGID   DYNAMICS. 


r   ! 


6.  A  rod  of  length  2  a  falls  from  a  vertical  position,  being 
capable  of  motion  about  one  end  in  a  vertical  plane,  and  when 
in  a  horizontal  position,  strikes  a  fixed  obstacle  at  a  given  dis- 
tance from  the  end.  Find  the  magnitude  of  the  impulse,  and 
the  pressure  on  the  fixed  end. 


Fig.  37. 


Let  the  rod  (Fig.  2i7)  <^^^op  from  the  vertical  position  and 
strike  an  obstacle  when  in  the  position  OB  with  a  blow  Q. 
Let  R  be  the  impulse  on  the  fixed  end  O,  and  then  we  have, 
taking  moments  about  O, 


Q- 


3 
■ — 7/ — ' 


and  since  the  rod  falls  from  the  vertical  position,  its  angular 
velocity  when  in  the  horizontal  position  is  found  in  the  usual 
way  to  be  given  by 

2a 

■•  ^=    d-^   3  ' 

The  impulsive  pressure  on  the  fixed  end  is  obtained  from  the 

relation 

Q  =  R  +  ^  iinro))  =R-^  Maw. 


MUTIUN   AliOUT   A    FIXED   AXIS.     IMl'ULSIVE   FORCES.     81 

^  2  <d  3  </      J 

These  two  values  of  Q  and  R  will  change  as  d  changes  ;  Q 

will  be  a  ma.ximum  when  (/=—-,  ;uul  A'  will  be  positive,  zero, 
or  negative,  according  as 

4  rt  >  =  <  3  ^, 


or  as 


Q. 


4a 


Hence,  if  the  obstacle  is  beyond  the  centre  of  percussion,  the 
impulsive  strain  at  O  is  downwards.  If  at  the  centre  of  per- 
cussion  there   is   no  impulsive  action   on  the  a.\is,   and  when 

d  <il-,  the  impulse  at  O  is  upwards. 
3 

These  results  can  easily  be  verified  by  experiment.  An  iron 
bar  movable  about  an  axis  in  which  it  is  very  loosely  held,  if 
dropped  so  that  it  strikes  an  obstacle  in  a  horizontal  position, 
will  throw  its  fixed  end  downwards  or  upwards  according  as  the 
obstacle  is  beyond  or  nearer  than  the  centre  of  percussion  ;  and 
if  the  bar  falls  so  that  it  strikes  the  obstacle  just  at  the  centre 
of  percussion,  then  there  is  no  jar  on  the  fixed  end,  no  matter 
how  loosely  it  may  be  held.  The  experiment  may  be  modified 
in  many  ways,  and  a  familiar  illustration  of  there  being  a  centre 
of  percussion  is  afforded  by  the  use  of  the  cricket  bat  or  base- 
ball club  with  which  a  ball  is  struck.  If  the  ball  be  struck  by  a 
portion  of  the  bat  out  near  the  end,  the  fingers  tingle  from  the 
impulsive  reaction  outwards ;  if  it  be  struck  nearer  than  the 
centre  of  percussion,  the  impulsive  reaction  is  inwards  against 
the  palm  of  the  hand  ;  when  the  ball  is  struck  properly,  there  is 
no  impulsive  reaction  on  the  hand,  and  the  energy  is  all  com- 
municated to  the  ball. 


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82 


RIGID   DYfJAMICS. 


7.  A  rod  is  moving  with  uniform  angular  velocity  aboat  one 
end  fixed  ;  suddenly  this  end  is  freed  and  the  other  end  fixed. 
Find  the  new  angular  velocity. 


O 


O' 


T 


.2L. 


O 


/' 


O' 


Fig.  38. 


1    . 


Fig.  38  indicates  the  solution.  In  the  first  figure  each 
particle  has  a  linear  velocity  wr  in  the  direction  indicated,  on 
account  of  the  angular  velocity  w.  In  the  second  figure  both 
ends  are  free,  and  the  velocities  remain  as  before.  In  the  third 
figure  O'  is  instantaneously  fixed,  which  does  not  affect  the 
velocities  of  tne  other  elements  of  the  rod,  by  the  definition  of 
an  impulse.  And  hence  w',  the  new  angular  velocity  about  0\ 
will  be  as  shown  in  the  figure  in  direction,  and  its  magnitude 
will  be  found  by  using  the  formula  for  moment  of  momentum. 
Thus 

And  if  .v  +  r=a,  and  p  is  the  density, 

.  •.  60  I    p(a  —x)x(Lv=M  •  —  G)', 

*/o  3 

. '.    to'  =  ),  (O. 


8.  A  rod  of  length  a  is  moving  about  one  end  fixed  with  uni- 
form angular  velocity,  when  sud(lenly  this  end  is  freed,  and  a 
point  distant  /  from  it  is  fixed.  What  in  general  will  be  the 
direction  and  magnitude  of  the  new  angular  velocity  .-* 

This  is  an  extension  of  the  preceding  i^roblcm,  and  the 
method  of  solution  will  be  similar.  Let  O  (Fig.  39)  be  the  first 
point  fixed,  and  the  angular  velocity  be  co,  as  indicated.  Then 
this  point  being  freed,  let  the  second  point  O'  be  fixed. 


MOTION   ABOUT   A    FIXED   AXIS.      IMPULSIVE    FORCES       8^ 

The  new  angular  velocity  will  be  obtained  by  equating  the 
moments  of  momentum  before  and  after  the  fixing  of  the  point 
0'.     Thus 

p  \    (ox{l-x)  .  dx-p  f     a)x{/+x)  ■  (ir=AU'^  (about  O')  x  &>'. 

For,  the  linear  velocity  of  an  element  at  P  is  {/-x)(o  before 
O'  is  fixed,  and  its  moment  of  momentum  about  O'  will  there- 


A 


A 


jr 


71 


I 


« 


O'       JP 


o 


\ 


Fig.  39. 


fore  be  vi{l—x)(xi'X\  while  the  moment  of  momentum  of  an 
element  at  Q  will  be  vi{l-\-x)w  •  x  in  an  opposite  direction  to  the 
former  with  reference  to  the  point  0\  If  p  be  the  density  and 
a  the  length  of  the  rod,  we  then  get  the  above  relation  which 
determines  the  sign  and  value  of  6)'. 

It  will  be  found  on  integrating  the  above  expressions  that  w' 
will  have  the  same  sign  as  co,  the  opposite  sign,  or  will  be  zero, 
according  as 

which  shows  that  if  a  rod  be  moving  about  an  axis,  and  this 
axis  be  freed  and  a  new  axis  fixed  through  the  centre  of  percus- 
sion, it  will  be  reduced  to  rest. 


I' 

I' 

HI 


9.  An  elliptic  lamina  is  rotating  with  uniform  angular 
velocity  about  one  latus  rectum,  when  suddenly  the  axis  is 
freed  and  the  other  latus  rectum  fixed ;  find  the  new  angular 
velocity. 


w  = ^—,w. 

1+4  e'' 


84 


RIGID   DYNAMICS. 


l    3 

'I; 
.1 


10.  A  circular  plate  rotates  about  an  axis  through  its  centre 
perpendicular  to  its  plane  with  uniform  angular  velocity.  If 
this  axis  be  freed,  and  a  point  in  the  circumference  of  the  plate 
be  fixed,  find  the  new  angular  velocity. 

Fig.  40  gives  the  solution.  For  an  element  at  P  the  linear 
^•elocity  is  <u  x  OP,  and  its  moment  of  momentum  about  O'  is 
w(ox  OPxO'P.  If  0/^  =  r  and  the  radius  of  the  plate  be  a, 
then  will 

2;;/a)/'(;-  — c?  cos  O)=jlf/v^(o'. 
.'.  Oft)  C  C%'Hr-acose)drde  =  M^—<o'. 


ftj 


ft). 


Fig.  40. 


II.    A  circular  plate   is   turning   in   its  own  plane  about  a 
point  A   on  its  circumference.       Suddenly  A    is  freed,  and   a 


t  a 
a 


MOT10x\   ABOUT   A   FIXED   AXIS.      IMPULSIVE   FORCES.      85 

point  /),  also  on  the  circumference,  is  fixed.  Show  that  the 
plate  will  be  reduced  to  rest  if  AB  be  one-third  of  the  cir- 
cumference. 

12.  A  triangular  plate  ABC,  right-angled  at  C,  is  rotating 
about  AC.  li  AC  hQ  loosed  suddenly,  and  BC  fixed,  find  the 
new  angular  velocity. 

,      BC 

G)    = 0). 

2  AC 

13.  A  square  lamina  is  rotating  with  angular  velocity  tu  about 
a  diagonal,  when  suddenly  the  diagonal  is  freed  and  one  of  the 
angular  points  not  in  the  diagonal  becomes  fixed  ;  prove  that 
the  angular  velocity  about  this  angular  point  will  be  I  o). 

14.  A  cube  is  rotating  with  angular  velocity  tw  about  a 
diagonal,  when  suddenly  the  diagonal  is  freed  and  one  of  the 
edges  which  does  not  meet  that  diagonal  becomes  fixed  ;  prove 
that  the  angular  velocity  about  this  edge  will  be   ^^  (o  V3. 

15.  A  uniform  string  hangs  at  rest  over  a  smooth  peg.  If 
half  the  string  on  one  side  be  cut  off,  show  that  the  pressure  on 
the  peg  is  instantaneously  reduced  by  one-third. 

52.    T/ie  Ballistic  Pcnduhim. 

This  is  a  device  for  measuring  the  velocity  of  discharge  of 
a  rifle  bullet,  and  was  invented  by  Robins  about  1743,  and 
afterwards  used  by  Dr.  Hutton  ;  and  although  of  recent  years 
superseded  by  the  more  accurate  electric  chronograph,  it  is 
to  be  noticed  here  as  illustrating  the  nature  of  an  impulse. 
In  its  simplest  form  it  is  a  heavy  pendulum  capable  of  moving 
about  a  horizontal  axis  ;  a  bullet  discharged  into  it  produces  a 
certain  angular  velocity,  and  the  pendulum  rises  through  an 
angle  which  can  be  easily  measured  ;  or  else  a  rifle  is  attached 
to  it,  and  the  discharge  of  the  bullet  produces  a  recoil. 

The  latter  method  is  shown  by  Fig.  41,  in  which  OA  repre- 
sents the  pendulum,  holding  the  rifle,  and  in   its  position  of 


I  I 


86 


RIGID   DYNAMICS. 


equilibrium  under  the  action  of  gravity.  The  bullet  being 
driven  out  produces  a  recoil  through  the  angle  «,  and  the 
velocity  of  discharge  is  found  as  follows  : 

Let  w/  =  mass  of  bullet, 

7'=  its  initial  velocity, 
/  =  distance  of  gun  from  O, 
iT/  =  mass  of  pendulum  and  gun, 
/'  =  radius  of  gyration  about  O, 
//  =  distance  of  centre  of  inertia  from  O. 

where  w  is  the  angular  velocity  generated. 


fi 


l!  I- 


77t^ 


The  pendulum  then  moves  back  through  the  angle  «,  which 
is  observed,  and  its  equation  of  motion  on  the  way  up  is 

d(o_  _gJi  sin^ 
dt~       IP-^-B 


MOTION   AIJOL'T   A   KIXEU    AXIS.     IMPULSIVK   FORCES. 


«7 


2  6)«&)  =  —  T-j-  -77,  I      %\XiQ 

U  It"  +  A'"*/!! 


^/^. 


.-.  6)2  =  7^^,(1 -cos «), 

and  this,  combined  with  the  previous  relation,  determines  v. 

In  the  other  method  a  similar  relation  will  be  found  ;  the  only 
difference  being  that  at  each  shot  the  pendulum  is  increased  in 
mass  by  the  addition  of  the  bullet  fired  into  it. 

A  rough  pendulum  made  of  a  wooden  bo.x  filled  with  sand, 
and  attached  to  an  iron  bar  which  carries  knife  edges  resting 
on  a  horizontal  smooth  plate,  will  readily  illustrate  the  above 
equations. 

The  preceding  solution  assumes  that  the  recoil  of  the  pen- 
dulum, when  the  gun  s  fired  without  a  ball,  is  so  small  that 
it  may  be  neglected.  Experiments  have  shown  that  this  as- 
sumption may  safely  be  made  for  small  charges  of  powder  but 
not  for  large  charges.  In  the  case  of  the  latter,  Hutton  as- 
sumed that  the  effect  of  the  charge  of  powder  on  the  recoil 
is  the  same  when  the  gun  is  fired  with  a  ball  as  it  is  when  it 
is  fired  without  a  ball.  Consequently  if  the  recoil  is  through 
an  angle  /3  when  the  gun  is  discharged  without  a  ball,  and 
through  an  angle  «  when  it  is  discharged  with  a  ball,  the 
velocity  of  the  ball  will  be 


Ml  \         2  2 


It   has   been   found    that   the   actual  velocity  of   the  ball    lies 
between  the  velocities  given  by  the  two  solutions. 


CHAPTER   VI. 


MOTION    ABOUT   A    f^IXED   POINT. 


ll!      I- 


•I 


\f^ 


Finite  Forces. 

53.  If  a  body,  fixed  at  one  point  only,  moves  under  the  action 
of  any  finite  forces,  then  at  every  instant  there  is  a  line  of 
particles  at  rest,  so  that  the  body  is  moving  about  what  is 
called  an  instantaneous  axis  passing  through  the  fixed  jjoint. 
Each  particle  will  have  a  certain  angular  velocity  about  this 
axis,  and  the  equations  of  motion  with  reference  to  any  three 
rectangular  axes  passing  through  the  fixed  point  can  be  written 
down  in  accordance  with  the  principles  already  explained.  In 
these  equations  the  expressions  for  thi  effective  forces  will 
have  to  be  evaluated  in  terms  of  the  angular  velocity  about 
the  instantaneous  axis,  and  in  order  to  explain  how  this  may 
be  done,  the  following  propositions  on  the  composition  and  reso- 
lution of  angular  velocities  will  be  found  useful. 


n 


54.    Angular  velocity  is    measured   in   the   same  manner   as 

linear  velocity  :   by  the  angle  described   in  a  unit  of  time  if 

10 
the  motion  be  uniform,  or  by  '      if  the  motion  be  not  uniform. 

(it 

It  may  be  represented  by  a  straight  line  drawn  in  the  proper 
direction,  and  perpendicular  to  the  plane  of  rotation.  And  it 
will  be  seen  that  angular  velocities  can  be  compounded  or 
resolved  in  the  same  way  as  forces  acting  at  a  point. 

Proposition   i.  —  For  angular  velocities  about  the  same  rota- 

tion  axis,  the  resultant  is  the  algebraic  sum. 

88 


I 


MOTION   AI50UT   A    FIXED   I'OINT. 


89 


This  is  evident,  since  the  successive  displacements  in  a  small 
time  are  superimposed. 

Prof'osition  2.  — //a  body  have  at  any  instant  two  angular 
velocities  about  tivo  axes  drawn  from  a  point,  and  if  lengths  OA, 
OB  be  taken  upon  the  axes  to  represent  in  direction  and  in  niao-. 
nitude  the  angular  velocities,  then  the  resultant  angular  velocity 
will  be  the  diagonal  OC  of  the  parallelogram  of  which  OA,  O/i 
are  adjacent  sides. 

Let  a  body,  fixed  at  O,  have  two  angular  velocities  repre- 
sented in  direction  and  in  magnitude  by  0^i,  0B\  and  let  the 
positive  direction  of  rotation  be  with  the  hands  of  a  watch. 
Take  any  point  P  in  the  plane  containing  OA,  OB,  and  con- 
.struct  Fig.  42.     And  let  6>/l=&)„,  OB  =  (o,,. 


Fig.  42. 

Then,  owing  to  &),.,  the  point  P  would  be  displaced  down- 
wards in  an  infinitely  small  time  dt,  a  distance  coa-PA/.dt 
or  (o^y  sin  AOB  •  dt.  Due  to  Wj  its  displacement  would  be 
upwards  (above  the  plane  of  the  paper)  and  equal  to  o)  PAWt 
or  Q)iX  sin  AOB  ■  dt. 

Therefore  the  total  displacement  of  P  is 

sin  A OB{y(i)^—X(o)di, 


m 


and  this  is  zero  when 


RICH)    DYNAMICS. 


.r 

y 

-t             V 

or           =  -   -• 

(O,, 

0) 

OA     OB 

ii  * 


';.!  -li 


which  is  the  equation  of  the  straii;ht  Hnc  OC.  And  thus  for 
all  points  along  OC  there  is  no  displacement  ;  that  is,  the  body 
is  turning  about  OC,  due  to  rotations  about  OA  and  OB. 

That  the  line  OC  represents  the  magnitude  of  the  resultant 
angular  velocity  may  be  shown  by  considering  the  displace- 
ment of  the  point  A.  Let  w,  be  the  resultant  angular  velocity 
about  OC. 

The  displacement  of  A,  due  to  w^,  is  zero. 

The  displacement  of  A,  due  to  wj,  is  OA  sin  AOP  •  <w///. 

The  displacement  of  A,  due  to  w^,  is  OA  ^\r\AOC-  u>Jt, 
and  therefore 

OA^\v\AOC  •  w,dt  =  OA  sin  A  OB  •  oj^df 

,.p   ii'mAOB      .-.^ 
sm  AOC 

Proposition  3,  —  ^/  ^  body  fixed  a!  a  point  have  anovular 
velocities  o)^,  oiy,  oi,  coinmnnieated  to  it  about  three  rectangular 
axes  passing  tlirough  the  fixed  point,  the  resultant  angular  veloc- 
ity is  given  by 

Also,  if  a  body  have  an  angular  velocity  w  about  an  instanta- 
neous axis  it  may  be  said  to  have  three  angular  velocities  w,,  w^, 
(1)^  about  three  rectangular  axes  ;  and  if  u,  /3,  7  be  the  angles 
which  the  instantaneous  axis  makes  with  the  coordinate  axes. 


then 
and 


(Or 


(O,. 


w 


cos  «     cos  j3     cos  7 


=  &), 


X 

ft). 


y 


fi>. 


give  the  equations  of  the  instantaneous  axis  when  w,,  eo^,  tu,  are 
known. 


^i^ 


MOTION    ABOUT    A    FIXKD    POINT. 


9« 


55.  That  a  poi»it  may  have  at  the  same  instant  three  angular 
velocities  can  be  seen  by  means  of  the  apparatus  shown  in 
Fig.  43- 


Fig.  4  3. 


are 


To  an  upright  stand  is  attached  by  means  of  pivots  a  system 
ol  two  rings  and  a  sphere.  The  outer  ring  can  rotate  about 
an  axis  passing  through  the  points  ^l,  />  ;  the  second  ring  may 
be  made  to  rotate  about  CD ;  and  the  inner  sphere  about  JiF. 

Now,  the  axis  AB  is  initially  in  a  horizontal  position,  and 
coincident  with  the  axis  of  x  drawn  from  O,  the  centre  of  the 
sphere ;  and  if  CD  be  made  coincident  with  the  axis  of  y  by 
placing  the  plane  of  the  outer  ring  in  the  plane  of  xj',  then 
it  is  evident  that  by  turning  the  inner  ring  the  axis  £F  may 
be  made  initially  coincident  with  the  axis  of  3. 

This  having  been  done,  rotations  may  be  given  first  to  the 
sphere,  then  to  the  inner  ring,  and  lastly  to  the  outer  ring; 
and  thus  any  point  on  the  sphere  will  have  simultaneously  the 


ij2 


KIt.lD    IJVNAMICS. 


f  ' 


■'  » 


three  anj^iilar  velocities  ^iven  to  the  system,  and  the  sphere 
will  rotate  about  a  result. int  axis  in  space,  which  would  be 
lixed  were  there  no  friction  at  the  pivots  ami  no  resistance 
of  the  air. 

The  arrangement  also  siiovvs  how  a  heavy  boily  may  be  fixed 
at  its  centre  of  gravity  and  at  the  same  time  be  given  rotations 
about  axes  fixed  in  sjjace. 

56.    Li  mar  Velocity  ixnd  Auij^nlar  Velocity. 

In  the  case  of  a  body  moving  with  one  point  fixed  we  may 
replace  the  angular  velocity  co  about  the  instantaneous  axis 
by  ft),,  w^,  (o,  about  three  rectangular  axes  drawn  through  the 
fixed  point.  The  next  thing  to  be  done  is  to  connect  the 
expressions  for  the  effective  forces  with  these  component 
angular  velocities  and  the  coonlinates  of  any  element  of  the 
body,  and    in  order  to  do  this  we  must   obtain   an   expression 

for  the  linear  velocities  '-^,  '-^, '  f  of  any  element  at  the  point 

dt     lit    dt  ^  ' 

(.r,  y,  d)  in  terms  of  ,v,  y,  a,  and  gj,,  w^,  oj,  ;  on  differentiating 

these  expressions,  we  shall  then  obtain  the  linear  accelerations. 

We  may  proceed  either  geometrically  or  by  direct  analysis. 


y  ■<  i 


I.    By  Gco))ictrical  Displacctticnt. 

Fig.  44  shows  how  the  linear  displacements  arise  from  the 
rotations  about  the  coordinate  axes. 

In  the  first  figure  the  body  is  supposed  to  be  fixed  at  O, 
and  01  is  the  instantaneous  axis  about  which  the  boily  is 
moving  with  angular  velocity  w.  The  body  may  be  supposed 
to  have  three  rotations  &)^,  w^,  (o,  about  the  three  coordinate 
axes  instead  of  ay  about  the  instantaneous  axis.  Then,  con- 
sidering positive  rotations  as  those  in  the  direction  of  the 
motion  of  the  hands  of  a  watch,  and  taking  the  displacements 
of  the  point  P{x,  y,  ::)  due  to  a  rotation  w„  we  have,  in  the 
second  figure,  P  moving  along  a  small  arc  PQ  in  time  dt,  due 


MOTION   AIIOUT   A    Kl,\i:i)    I'UINr. 


03 


to  ft),,     This  small  (lisplaccnu-nt  /'(J  is  f(|uivaleiU  to  two  J'K, 
RQ  in  the  directions  indicated.      Ilciuc  \vc  have, 


and 


Fig.  44. 

And,  by  considering  the  other  planes,  we  should  get  the  dis- 
placements of  P  due  to  ft),  and  to  <>',,  thus  : 

Along  Ox  Oy  Oz 

Displacements  due  to  o),     —yw^dt        xw^dt 

Displacements  due  to  tu,  —zoa^dt        y/odt 

Displacements  due  to  w^        zw^dt  -xw^dt 

These  are  written  down  symmetrically  ;  and  from  them  we 
see  that  the  linear  displacement  along  Ox,  which  we  call  dx,  is 


I  <   !| 


i^ 


H 


J,  I 


94 


RIGID   DYNAMICS. 


equal  to   {sot^—yoi^.dt,  and,  therefore,   in  the  limit  the  linear 
velocity 


dt 


=  z(o^-y(0,. 


dy 
dt^'^'^'~"^'* 


and 


d^ 
dt 


^yco,-XQ)^. 


2.    By  Direct  Analysis. 

Let  the  body  (Fig.  45)  be  fixed  at  the  point  O,  and  let  01  be 
the  instantaneous  axis  as  before,  and  the  angular  velocity  ro  be 


fi. 


\A 


.'I     i 


i\ 


)     I 


Fig.  45. 


equivalent  to  w,,  m^,  «„  as  shown.     Then  an  element  at  P  is 
tending  to  move  at  any  instant  in  a  circle  about  (9/,  and  its 


be  linear 


et  01  be 
city  to  be 


/ 


MOTION   ABOUT   A   FIXED   POINT. 


lis 


95 


absolute  velocity  is  wp  —  -~,  where/'  is  the  perpendicular  from  P 

on  the  instantaneous  axis. 

And,    if   «,  /S,  7  be   the   angles  which    01  makes  with    the 
coordinate  axes,  then 

f^={pzQ^^—y  cos  7)2 H h  •••  ; 

also       ^,  -^,  —  are  the  direction  cosines  of  the  tangent  at  P, 
ds  ds  ds 

-,  •^,  "   are  the  direction  cosines  of  OP, 
r  r  r 

cos  «,  cos  /8,  cos  7  are  the  direction  cosines  of  01. 

And,  since  OP  is  perpendicular  to  the  tangent  at  P,  and  01 
also  perpendicular  to  this  tangent,  we  have 

dx    X  .  dy   y  ,  dz    z     ^ 
ds     r     ds     r     ds    r 


dx 


dv 


dz 


ax            ,  nv         a  I  "^  ^ 

—  cos  «  +  -^  COS  l3-\ cos  7  =  o. 

ds  ds  ds 


r 


dx 

ds 


dy 

ds 


dz 
ds 


z  COS  ^—y  cos  7     x  cos  ^^  —  z  cos  «    y  cos  «  —x  cos  /3    / 

ds 
And,   therefore,    since  —  =  «A   we   have,    multiplying    each 

dt 
quantity  by  --> 


dx 


=  {z  cos  yS  —J  cos  7) ft)  =  rft>„  — ji'ft)^, 


:  at  /*  is 
',  and  its 


dt 

dz  _ 
'dt~ 


as  found  betore. 


anal, 
anal. 


--ao)^  — ^ft),, 


-.yw^-xojy, 


96 


RIGID    DYNAMICS. 


11 


fi-' 

■i; 


!^1: 


57.  The  former  of  the  two  investigations  in  the  preceiling 
article  may  be  presented  in  purely  analytical  form  thus  : 

(i)  From  the  point  P  (Fig.  45)  let  fall  perpendiculars  on  the 
coordinate  axes  OX,  OV,  OZ,  and  let  9,  <f>,  yfr  be  the  angles 
which  these  perpendiculars  make  with  the  coordinate  planes 
XV,    YZ,  ZX.     The  angular  velocity  of  P  about  the  axis  OX 

will  be  — -,  and  the  resolved  parts  of  this  parallel  to  the  coor- 
dt 


dinate  axes 


Now 


.„  ,      (bx\de    fdv\dd        ,   fdz\de  ,.     , 


and 


y  =  ^{>^—x^)  cos,  6, 

z-=^{i'^—x^)  sin^, 

bx 
dd 


=0. 


and 


And  by  definition, 


80' 

83 
89 


^/{r^—x^)  sin  6=  —z, 


=  V  (r^  —  x^)  cos  6  —y. 


dd 
dt 

'8x\de 


=&), 


(8x\de 


and 


(: 


'^_z\d±^ 


89)  dt 


r<Wx 


Treating  the  rotations  about  (9Fand  OZ  in  like  manner,  we 
obtain  the  following  complete  system  of  equations : 

x=  ^{>'^~z^)  cos  i/r=  y(/^-j'2)  sin  ^, 

y=  V('''^--0  cos  9=  ^(;'2_y^)  sin  1^, 

z=  V(r2_y2)  cos  <^.=  y(r2  -.1-2)  sin  9  : 


I 


MOTIOxN   ABOUT   A    FIXED    POINT. 

(dx\dd  (Oy\!e__        fd.:r\<W   ^ 

^(f>Jdt      '  \d(\>)dt       '  ''  \d(l>jdt 


97 


#  ._.. 


3£\^_         /  dx  \dyfr  _ 
~~        ~   '     .dfjdt  ~ 


(  ds  \d'\}r 
\d^J  dt 


-y(o. 


dfj  dt 


The  total  velocity  parallel  to  OX  is  the  al<;cbraic  sum  of  the 
partial  velocities,  that  is 

dx  ^  (bx\ie     (dx  Y4>     f  dx\d± 

dt    \ddjdt    \o'4>rdt    V)fjdt' 

dx       dd>       d-^ 


Similarly, 
and 


dy       d-^       dO 
dt        dt       dt 


ds       dd       d(f> 
dt       dt        (it 


(2)    The  second  investigation    in   Art.   56  may  also  be  pre- 
sented in  a  purely  analytical  form  thus  : 

,^^2+y+.2^;.2^  (I) 

.i-cos«4-j/ cos/3  +  ,:r  cos  7  =  »'Cose,  (c  =  angle /(9/^), 

/)2=  {p  cos  /3-j/  cos  7)2+  (.r  cos  7 -.7  cos  a)'^  +  {y  cos  a-,r  cos  7)"'^. 


Also 
and 

and 


«o. 


"^JL-^^^^n, 


cos  a     cos  y8     cos  7 
.  •.  xci)^  +  J'ft>»  +  -<«,  =  ^'«  cos  e, 


(2) 


(fT-<fT+(fJ=<-'--'->^+<-">^+*-^--^-"'''^-'^' 


H 


l>'  .1 


:U 


If;.' 
(; 


98 


RIGID    DYNAMICS. 


The  body  being  rigid  and   O  a  fixed  point  in  it,  r  and  e  are 

constants  ;   also  w,  o)„  w^,  a>,  are  independent  of  x,  jy,  a,  the 

coordinates  of   P,  therefore    from   (i)  and   (2)  we  obtain,    by 
differentiation. 


dx  ,    dv  ,    dz 
dt       dt      dt 

dx  ,      dy  ,      d:; 


dx; 

dt 


dy 
dt 


da 
dt 


:;a)^  —ya^     .rco,  —  aa^    yw^  —  xco^ 


=  ±  ^  by   (3). 


(4) 


The  ambiguity  of  sign  in  (4)  arises  from  the  fact  that  in 
equations  (2)  and  (3)  there  is  i  othing  to  determine  whether  the 
rotations  to,,  (o^,  cd,,  are  in  the  directions  x  to  y,  y  to  ;y,  z  to  x,  or 
in  the  opposite  directions  x  to  ^,  j:  to  y,  y  to  x.  If  they  are  in 
the  former  directions,  the  value  + 1  must  be  taken,  if  they  are 
in  the  latter  directions,  the  value  —  i  must  be  taken. 

58.    General  Equations  of  Motion. 

Let  the  body  be  in  motion,  with  one  point  O  (Fig.  46)  fixed, 
and  let  three  rectangular  axes  be  drawn  from  O,  OX,  OY,  OZ^ 
to  which  we  may  refer  the  position  of  the  body  at  any  time  dur- 
ing its  motion.  And  let  it  be  acted  upon  by  external  forces, 
producing  on  each  element  of  the  body  m,  accelerations  X,  V,  Z 
in  the  directions  of  the  three  fixed  axes.  Then,  if  P  be  the 
pressure  on  the  fixed  point,  and  X,  /li,  v,  the  angles  which  the 
direction  of  the  pressure  makes  with  the  fixed  axes,  we  have, 
by  D'Alembert's  principle,  the  relations 


2;//  — '^  =  Sw A'  +  P  cos  \, 
dr 

2;//  -j-^=1;n  Y  -\-  P  cos  fi, 

2;// — ^=1mZ  +Pcosu. 
dt^ 


(I) 
(2) 

(3) 


MOTION   ABOUT   A  FIXED   POINT. 


99 


And,  also, 


(4) 


S//^ 


d^y\ 


Im 


d\v       d'h 


:;;/  .x 


dfi 
d^V 

^'d^-y 


d^x\ 
dt'^J 


(4) 
(5) 
(6) 


Fig.  46. 

where  L,  M,  A^are  the  couples  due  to  the  external  forces.  Now, 
since  the  body,  when  we  form  the  above  equations,  is  moving 
about  an  instantaneous  axis  01  with  some  angular  velocity  <u, 
which  we  may  suppose  equivalent  to  a),,  w^,  co„  and  since  we 

have  shown  that  ^~-  =  sii)  —yw.,  -■^—.vto^  —  so)^,  and  -^=^&),— ;tra)^, 
dt  '    dt  dt 


l:| 


lOO 


RIGID    DYNAMICS. 


■I 

ii  i 


it  is  evident  that  these  equations  can  be  expressed  in  terms  of 
known  quantities,  and  o),,  (o^,  to,. 

Hence  the  equations  (4),  (5),  (6),  taken  with  the  initial  cir- 
cumstances, will  serve  to  determine  w^,  co^,  w,,  and  therefore  w 
and  the  position  of  the  instantaneous  axis;  equations  (i),  (2), 
(3)  will  then  give,  on  substitution,  the  value  of  P. 


f  1 


1 1 

i  , 


'  I  . 


59.    Equations  of  Motion  referred  to  Axes  fixed  in  Space. 

Taking  the  equation  (6),  we  shall  proceed  to  evaluate  it  in 
terms  of  <u^,  w^,  to,  by  taking  the  values 


dx 


dv 


dz 


and  differentiating. 

Thus  we  should  get  -7-.,  =  ^    ,   4-&)„-r  — J^   v^  — <w--v-.  and  on 
'^       dt^         dt        "  dt    -^  dt        -  dt 

substituting  in  this  the  values  of  -^,    -j-,  we  get 

d^x        da)„        d(o^       /    a  1      2  ,      2\  ,       /         ,  ,         \ 

"^"^  '^  "^"-^  ^~'^'^  '  '^'^     -^'"''       '"'^' 

d'h-        da>         d(o,       o     ,      /         ,  ,        x 

'dfi^^dt~^'di~  '""''*'     <»A-^«x  +J'&>,  +  ^&),), 

and,  similarly,  we  should  get,  by  symmetry, 

d"^)!        da),       d(o^       ^ 

~dt'^  "■^'  dt  ~    ~dt~  ^^'     '^^^^''^'^  ^y^^  +  ■^^'"')- 

Therefore  relation  (6)  becomes 
Ar    V     (   dh        d\v\ 

=  2,  /« {x"^  -^y^)  '  — -  —  2  7nxs  — -  —  2  myc  — »f 
dt  dt  dt 


it'. 


MOTION    ABOUT   A   FIXED   POINT. 


lOI 


IS  of 

cir- 
rc  ft) 

(2), 


t  in 


and  by  analo;j;y  J/ and  L  can  be  written  down.  Since  L,  M,  .V 
are  given,  these  results  would  give  the  values  of  &>„  eB„,  to,  on 
integration,  after  calculation  of  the  moments  and  products  of 
inertia  required.  But  this  calculation,  as  can  be  seen,  would 
be  tedious,  and  we  can  avoid  most  of  it  by  choosing  axes  which 
although  still  fixed  in  space  are  in  coincidence  with  the  princi- 
pal axes  of  the  body  when  we  form  the  equations  of  motion. 
This  device  enables  us  at  once  to  disregard  the  products  of 
inertia,  and  makes  a  great  simplification  in  the  problem.  It  is 
due  to  Elder,  and  the  equations  thus  obtained  are  known  as 
Elder  s  equations  of  motion. 


1  on 


60.    Elder  s  Equations  of  Motion, 

Instead  of  choosing  any  three  rectangular  axes  fixed  in  space 
at  the  instant  under  consideration,  let  axes  be  so  chosen  that 
they  coincide  with  the  principal  axes  of  the  moving  body;  and 
let  ft)j,  ft)2,  0)3  be  the  angular  velocities  about  these  principal 
axes,  which  will  then  be  the  rame  as  <u^,  <Oy,  co^  in  the  preceding 
equations.     We  shall  then  have 


■       dt  -^       ^ 


ft), 


2' 


and  it  can  be  shown  (see  Art.  61),  that  -^=^'  '"3. 

^  '  dt       dt 

.-.    ;V^=2;;/(;tr2+y)'^^  +:^;«(.v2-/)ft)ift),. 

dt 

Thus  the  equations  for  determining  <o^,  w^,  6)3  become 


d(o< 

1 

dt 

^d(Oq 
'  dt 

,d(i)o 


A--^-(B-C)a>^(o^  =  L, 


B'^^^-{C-A)a>^<o,=^M, 


C'^]f^-{A-B)ro,co^=I^. 


I02 


RIGID   DYNAMICS. 


K 


riicse  equations,  on  integration,  being  three  in  number, 
should  serve  theoretically  to  determine  the  angular  velocities 
tyj,  0)2,  0)3,  and  the  position  of  the  instantaneous  axis.  The 
actual  situation  of  the  body  with  reference  to  known  direc- 
tions in  space  can  also  be  found  from  these,  combined  with 
certain  other  relations  which  will  be  given  further  on. 


Ml 


U 


61.    It  might  seem  that  -^/=    -3  follows  at  once  from  the 
relation  a),=G)3,  but  it  does  not  necessarily  so  follow;  that  the 


Fig.  47. 


former  relation  holds   as  well  as  the   latter   may   however   be 
shown  in  the  followintr  manner: 

Let  OX,  O  V,  OZ  be  the  three  axes  fixed  in  space  (Fig.  47) ; 


MOTION    AHOUT   A    KIXED   POINT. 


103 


then  a  body  movinj;-  with  the  point  O  fixed  will  produce  a])out 
OP  an  angular  velocity 

eoj,  cos  a  +  (Uj,  cos  y8  +  <u,  cos  7, 

if  a,  /3,  7  are  the  angles  which  01^  makes  with  the  axes. 

Differentiating  this  expression  with  respect  to  /,  we  get  for 
the  angular  acceleration  about  OP 

cos«^'-a,,sin«^  +  cosy8^"^-a,,sin^'-^  +  cos7^' 
dt  dt  lit       '  dt  '  di 


—  ft),  sin  7 


dt 


i    I 


Suppose  now  that  OP  approaches  OX  and  ultimately  coin- 
cides with  it,  then  the  angular  acceleration  becomes 

dw,         d^         dy 
dt        'dt         dt 


because  a=o,  /3=7  =  -  when   OP  coincides  with   OX:   and  it 

2 

d/3 


is  also  evident  from  the  figure  that  in  such  case  -^  is  the  same 

d  ^^ 

as  ft),  or  ft)q,  and  that  -^  - 

^  dt 


(Oy  or  —  G).2. 


-^  =  -— ^  at  the  same  time  that  &),  =  &>,. 
dt       dt  ^       ' 


The   relations    between    ^J,    ^,    -y\    the   angular    accel- 

dt       dt       dt 

erations   around   axes   fixed   in  the  body,  and   — ^    ^^    — ', 

dt       dt       dt 
the  angular  accelerations  around  axes  fixed  in  space,  may  be 

determined    for    any    given    position    of    the    moving    body, 
as  follows  : 

Let  /j,  Wp  «j ;  /g,  Wg,  «2  5  4>  '^^a*  ''3>  ^^  the  direction  cosines 


|5i 


1 1 


Ml 


104 


RK'.ID    DYNAiMICS. 


of  axes  fixed  in  the  body  referred  to  coordinate  axes  fixed  in 
space.     Then  will 


(U2  =   /aft),  4-  f^i^fOy  +  n^di. 


<Wx=  A<"i+  V^-j+  4<y3 


(I) 


(2) 


(3) 


0)^  = ;//  ,cy  J  4-  WgW^  +  ''/3W3 
/i^  + ;;/  ^J^ti^-=\,     l^I.,-\-  m  1  ni^  + ;/ 1;/2  =  o, 

i.^  +  ;//2^  +  "i  =  ' '       4^3  +  WoW/.j  +  //o/Zg  =  O, 

^z  +  ^''a'^  +  ''z  =  ' '     's'^i  +  WgW  1  +  ^/g/Zi  =  o. 
Differentiating  the  first  equation  in  group  (i), 

d(o^        (iw,         dw^        d(o,        dL  ,      dm.         dn, 

T?7  ='■;* +"''7ff +'''rfr+"v/+"'-,7r+"-7?F- 


But  the  sum  of  the  last  three  terms  on  the  right  hand  side  of 
this  equation  is  zero,  for 

dL        dm^        dfi, 

'''-dt''''^dr^''iu 


jh 


dm^ 


=  (/jO)!  +  4ft>,^  +  ''3<«3);77  +  (''^1®1  +  W2«2  +  ^"Z^^  ~Jl 


1  /         .  1         \^^h 


(  ,dL         dm-,  ,      duA  f  ,dL 


wr 


dnt]        dn.  \ 
^^''^dt) 


+ 


/ ,  dL         dm^        d)i\ 

v^-dj-^'"^  dt'-^'^'-dih 


=  0 


;d  in 


(I) 


(2) 


(3) 


le  of 


^ 


'•)«2 


(Oo 


MOTION    AIIOUT   A   FIXED   POINT.  105 

as  appears  at  once  on  diffcrentiatinf;  the  equations  in  groiij)  (3). 

(4) 


r/o). 


(/(o,       ,  (io),  ,._,         ..„, 


I". 


From  the  second  and  third  equations  in  group  (i)  \vc  may  in 
liicc  manner  obtain 


and 


lit        ^  dt  ^(it         ^  lit 


''  dt ' 


Hence  the  acceleration  around  any  axis  may  be  projecteil  on 
coordinate  axes  just  as  angular  velocities  and  as  segments  of 
the  rotation  axis  may  be  projected,  and  all  theorems  on  the 
projection  of  segments  of  a  line  may  be  interpreted  as  theorems 
on  the  projection  of  angular  accelerations  about  the  line. 

If  the  axis  of  to^  coincide  at  any  moment  with  the  axis  of  tw^, 
then  will  /i=  i,  ?;/i  =  o,  «i  =  o,  a>i  =  a),,  and  by  (4)  above 

dw^_d(ii^ 


dt 


dt 


62.    Angular  Coordinates  of  the  Body. 

The  equations  of  motion  known  as  Euler's  enable  us  to  find 
ft)j,  ft)2,  Wg,  the  angular  velocities  of  the  body  with  reference  to 
the  principal  axis  drawn  through  the  fixed  point  about  which 
the  body  is  moving.  As  these  principal  axes,  however,  are 
in  the  body,  and  move  with  it,  we  must  have  some  means  of 
determining  the  position  of  the  body  with  reference  to  axes 
fixed  in  space,  because  the  values  of  the  angular  velocities 
found  by  solving  Euler's  equations  tell  us  nothing  whatever 
as  yet  of  the  situation  of  the  body  with  regard  to  any  known 
directions  in  space.  In  order,  then,  to  fix  the  position  of  the 
body  at  any  time  and  give  us  a  definite  idea  of  its  situation 
with  reference  to  some  initial  position,  three  angles  6,  <f>,  -yjr 


':! 


u. ./ 


lit 


io6 


KUWU   DYNAMICS. 


cc^sirvf        ^sinS 


0)2 


COfCOSO 


Fig.  48. 


MOTION   AUOin"   A    IIXEIJ   I'OINT. 


107 


cos^ 


arc  chosen,  known  as  the  angular  coordinates  ;  they  define  the 
situation  of  the  i)rincii)al  axes,  and  therefore  of  the  body  itself, 
beinj;  measured  from  some  initial  fixed  axes  of  reference  which, 
at  the  i)ej;innin<;  of  the  motion,  coincide  with  the  principal  axes 
of  the  body.  Relations  can  be  easily  found  between  0,  0,  y^r, 
and  <u,,  0)2,  (Ug,  so  that  knowing  tiie  anj;uiar  velocities  we  can 
find  0,  <^,  \/r,  and  the  motion  of  the  body  is  fully  known.  'Ihc 
subjoined  fi};ures  (I''i};.  48  and  F*ig.  49)  show  how  the  position 
of  the  principal  axes  at  any  instant  may  be  determined  by 
displacements  0,  (f>,  yjr  \  they  also  indicate  how  the  relations 
existinj^  between  these  displacements  and  the  an<;ular  velocities 
about  the  principal  axes  are  to  be  found. 

Let  a  spherical  surface  of  radius  unity  be  constructed  at  the 
fixed  point  C)  (I'i^C-  4'^)»  about  which  we  suppose  a  body  to  be 
moving.  Initially,  let  the  body,  which  we  may  represent  by  its 
principal  axes  O/l,  OB,  OC,  be  in  such  a  position  that  OA,  (U\ 
t^C  coincide  with  OA',  OV,  0/  respectively.  Then,  by  suppos- 
ing the  body  to  turn  through  the  angles  i/r,  0,  ^  in  order,  so 
that  the  point  A  travels  in  the  directions  indicated  by  the 
arrows,  it  is  evident  that  af/y  position  of  the  body  will  be  fully 
known  in  respect  of  the  fixed  axes  0.\\  O  )\  OX,  when  we  know 
three  such  angles  as  6,  <^,  i/r. 

At  any  instant  the  body  has  angular  velocities  Wj,  &)„,  0)3  indi- 
cated by  arrows  ;  and  in  order  to  connect  these  with  the  angular 
coordinates,  consider  the  motion  of  a  particular  point  such  as  C. 
The  velocity  of  the  point  C  at  the  instant  in  question,  may  be 
considered  as  the  resultant  of  the  angular  velocities  co^,  Wo,  wg, 

or  as  due  to  changes  in  6,  (fy,  yjr,  i.e.,  to  velocities   '-—,  ^—  ,  ^  J-; 

at    at     at 

and  by  expressing  in  the  two  systems  of  change  the  velocity  of 

C  resolved   in  three  determinate  directions,  and  equating  the 

lesults,  we  shall  arrive  at  the  relations  between  (Up  Wg,  Wg,  and 

dp   d(f)   (i± 

lit'   dt'   dt' 


\ 


io8 


RIGID   DYNAMICS. 


til 


?!;'■ 


The  auxiliary  figure  shows  the  motion  of  the  point  C  due  to 
the  two  systems.  The  line  ZCZ^  is  the  tangent  to  the  line 
of  the  great  circle,  and  the  point  C  will  evidently  have  angular 
velocities  w^  co.^  in  the  directions  indicated  by  the  arrows  ;  it 

has  also  a  motion  —  along  the  tangent  to  the  great  ci  xle  at  C 

dt 

and    a   motion  —f-  sin  d   perpendicular   to   this    former.     This 
at 

velocity  ^  sin  6  arises  from  the  fact  that  C,  owinij:  to  the  -v|r 
lit  'or 

motion,  has  a  velocity  along  a  tangent  to  a  small   circle  with 
CC  as  radius,  and  its  velocity  perpendicular  to  ZCZ^  must  be 

CC  •  ^-^=OC^\x\.  6  •  — ^  =  --J^  sin  6,  since  we  have  agreed  to  call 
(U  dt      dt  *= 

the  radius  OC  unity. 

Hence,  we  have  from  the  auxiliary  figure,  remembering  that 
the  radius  is  unity,  the  relations, 


dQ 
velocity  of  C  along  ZC=—-  =  (i>^  sin  A  +  Wocos  6, 

dt 


(I) 


velocity  of  C  perpendicular  to  ZC='^^  sin  6 

dt 


^d± 
~  dt 
-  —  Wj  cos  (f>  +  o)2  sin  <f). 


(2) 


M:    1 


And  by  considering  the  motion  of  the  point  E,  we  have  the 
velocity  of  E  along  the  tangent  at  E  equal  to 


'^+0E  cos  0  .  ^==#  +  #:  cos  e  = 
dt  dt      dt      dt 


0)n 


(3) 


The  relations  (i),  (2),  (3),  along  with  Euler's  equations  of 
motion,  Art,  60,  give  a  complete  solution  of  the  problem  as  far 
as  the  actual  motion  and  position  of  the  body  are  concerned. 

Fig.  49  is  given  merely  to  show  how  the  principal  axes  which 
at  any  time  really  represent  the  body  itself  were  initially  coin- 
cident with  the  fixed  axes  in  space,  and  have  turned  through 
angles  6,  </>,  i/r.  The  complications  in  the  former  figure  are 
onitted. 


m 


(I 


MOTION   ABOUT   A   FIXED   POINT. 


109 


This 


(I) 


Fig.  49. 


63,    Pressure  on  the  Fixed  Point. 

The  pressures  on  the  fixed  point,  measured  along  three  fixed 
rectangular  axes,  will  be  given  by  the  equations, 


(3) 


^w^^SwA'+Z'cosX, 
dt^ 


11 


I 


2;«^  =  2wF+Pcos/ti, 
dt^ 


dt^ 


d^V 

where  2;;/—-  is  now  to  be  expressed  in  terms  of  the  coordi- 

dfi 
nates  of  the  centre  of  inertia,  the  mass  of  the  body,  and  the 


no 


RIGID   DYNAMICS. 


I, 


m 


angular  velocities.     Thus,  if  we  evaluate  as  formerly  — ^  in  terms 
of  0),,  a>y,  ft),,  we  get 


2;// 


ci^x 


dt^ 


—  Sw  -I , 


dt  dt 


G)2.r-f  GJx(^«x+/a>y+"a)J  \  ; 


and  if  x,  y,  z  be  the  coordinates  of  the  centre  of  inertia,  we 
have,  on  reduction,  to  determine  the  three  pressures. 

Mass  •]  ~5-^— j/^^  — &)2i'  +  to,(.rG)^+3'a>j,  +  5wJ  [  =/'cosA,  +  2w/A', 
l     dt         dt  ) 

and  two  similar  relations  for  P  cos  /a,  P  cos  v. 

These  equations  are  with  reference  to  axes  fixed  in  space ; 
but  if  we  refer  them  to  the  principal  axes  moving  with  the  body, 

we  may  use  Euler's  equations,  and  substitute  for  — >',  — ',  — - 

dt      dt     dt 

their  values  in  terms  of  A,  B,  C,  L,  M,  N,  w^,  co^,  co^. 
The  equations  when  finally  reduced  in  this  way  become 

Mass  .  |coi(^+6^-^)(^+^)-K2+ft)32)I-} 

=/'cos\4-2;«JSl'— Mass  •  ]77^"~7;j[' 

with  the  two  analogous  expressions  for  P  cos  fi,  P  cos  v.  In 
these  expressions  x,  j,  2  are  the  coordinates  of  the  centre  of 
inertia,  L,  M,  iVthe  couples  due  to  the  external  forces,  A^  B,  C 
the  principal  moments  at  the  fixed  point. 

And  it  is  evident  that  if  x—y='z  =  o,  the  pressure  on  the 
fixed  point  will  be  the  resultant  of  the  external  forces  ^mX, 
2/;/  F,  2///Z ;  as,  for  example,  in  the  case  of  a  heavy  body  fixed 
at  its  centre  of  gravity,  where  the  pressure  must  be  simply  the 
weight  of  the  body. 


iil 


MOTION   ABOUT   A   FIXED   POINT. 


Ill 


Illustrative  Examples. 

1.  If  cu,,  coj,,  w,  be  the  angular  velocities  about  the  coordinate 
axes  by  which  the  motion  of  a  body  about  the  origin  may  be 
exhibited,  find  the  locus  of  the  points  the  magnitude  of  whose 
velocity  is  aw^. 

2.  The  locus  of  points  in  a  body  (which  is  moving  with  one 
point  fixed)  that  have  at  any  proposed  instant  velocities  of  the 
same  magnitude,  is  a  circular  cylinder. 

3.  A  body  fixed  at  one  point  moves  so  that  its  angular 
velocities  about  its  principal  axes  are  a  sin  ;//,  a  cos ;//,  in  which 
t  represents  the  time,  and  n  and  a  are  constants.  Show  that  the 
instantaneous  axis  describes  a  circular  cone  in  the  body  with 
uniform  velocity. 

4.  A  uniform  rod,  length  2  a,  turns  freely  about  its  upper 
end,  which  is  fixed,  and  revolves  so  as  to  be  constantly  inclined 
at  an  angle  «  to  the  vertical.  Find  the  direction  and  magni- 
tude of  the  pressure  on  the  fixed  end. 

5.  Any  heavy  body,  for  which  the  momental  ellipsoid  at  the 
centre  of  inertia  is  a  sphere,  will,  if  fixed  at  its  centre  of  inertia, 
continue  to  revolve  about  any  axis  around  which  it  was  origi 
nally  put  in  motion. 

6.  A  right  circular  cone,  whose  altitude  is  equal  to  the  diam- 
eter of  its  base,  turns  about  its  centre  of  inertia,  which  is  fixed, 
and  is  originally  put  in  motion  about  an  axis  inclined  at  an 
angle  a  to  its  axis  of  figure.  Show  that  the  vertex  of  the  cone 
will  describe  a  circle  whose  radius  is  ^  a  sin  «,  a  being  the 
altitude. 

This  is  evident,  since  the  momental  ellipsoid  at  the  centre  of 
inertia  of  the  cone  is  a  sphere ;  therefore  the  cone  will  revolve 
about  the  original  axis  permanently  (Ex.  5  above),  and  its  axis 


,:'i,| 


112 


RIGID   DYNAMICS. 


will  describe  another  cone,  and  its  apex  will  trace  out  a  circle  of 


radius  |  «  sin  «. 


7.  A  circular  plate  revolves  about  its  centre  of  gravity  fixed. 
If  an  angular  velocity  qj  were  originally  impressed  upon  it  about 
an  axis  making  an  angle  a  with  its  plane,  show  that  a  normal 
to  the  plane  of  the  plate  will  make  a  revolution  in  space  in  time 


27r 


<uVi  +3  siu'^of 

8.  A  body  has  an  angular  velocity  w  about  a  line  passing 
through  the  point  «,  /3,  7,  and  having  direction  cosines  /,  w,  n. 
Show  that  the  motion  is  equivalent  to  rotations  /w,  7uco,  nco 
about  the  coordinate  axes  and  translations  {in'y  —  n^)a),  {na  —  /y)co, 
{lj3  —  inu)(t)  in  the  directions  of  these  axes. 

9.  A  body  has  equal  angular  velocities  about  two  axes  which 
neither  meet  nor  are  parallel.  Show  that  the  motion  is  equiva- 
lent to  a  translation  along  a  line  equally  inclined  to  the  two 
axes  and  a  rotation  about  this  line. 


f  ( 
Si  , 


64.    Top  spinning  on  a  Rough  Horizontal  Plane. 

When  a  common  top,  symmetrical  with  respect  lo  its  axis, 
is  spun  and  placed  on  a  rough  horizontal  plane,  with  its  axis 
inclined  at  an  angle  to  the  vertical,  it  satisfies  approximately 
the  conditions  Tor  motion  about  a  fixed  point ;  and  we  may 
first  consider  the  ideal  case  of  a  top,  spinning  on  a  perfectly 
rough  horizontal  plane,  with  its  apex  fixed,  and  free  to  move  in 
all  directions  about  this  apex  considered  as  a  fixed  point. 

Lr<"  a  top.  Fig.  50  (i),  be  set  spinning  about  its  axis,  and  placed 
on  a  rough  horizontal  plane,  with  its  axis  inclined  to  the  vertical 
at  a  given  angle.  Then  after  a  certain  time  its  position  with 
reference  to  fixed  lines  in  space  will  be  as  indicated  in  the 
figure  by  its  principal  axes  OA  OB,  OC,  drawn  through  the 
fixed  point.     G  is  the  centre  of  gravity  of  the  top,  and  OG  =  h. 


MOTION   ABOUT   A   FIXED   POINT. 


113 


of 


Fig.  50. 


i!l 


I 


114 


RIGID   DYNAMICS. 


The  angle  Z0C=6,  and  the  line  NON'   is  the  line  of  nodes. 

The  angular  velocity  ~  is  called  the  Nutation,  and  iX  the  Pn- 

dt  dt 

cession. 

The  top  is  acted  upon  only  by  the  external  force  of  gravity, 

since  we  suppose  an  ideal  case  first  and  neglect  the  couple  of 

friction  acting  at  the  fixed  point,  as  well  as  the  resistance  of 

the  air.     The  external  couple  is  equal  to  MgJi  sin  d,  as  is  seen 

from  Fig.  50  (3),  which  tends  to  turn  the  top  about  the  line  of 

nodes.     This  couple  may  evidently  be  resolved  into  two  others, 

one  equal  to  Mgli  sin  d  cos  <^,  tending  to  turn  the  top  about  OB, 

and  the  other,  equal  to  MgJi  sin  ^sin<;£),  tending  to  turn  it  about 

OA,  as  may  be  seen  from  Fig.  50  (2).     Hence  the  equations  of 

motion  are 


I'!        I 


W,  1 


A  —-^  —  (/?  —  C)(i>.^w^  =  Mgh  sin  6  sin  (^, 
dt 

B'^'^-{C-A)oi^(o^  =  Mghs,mecos<i>, 
dt 

C'^-(A-B)co,co^  =  0, 


(I) 
(2) 

(3) 


and  we  have  also  the  relations 


ill 


de 

dt 


=  (o^  sin  (f>  +  (02  cos  <{), 


■    — ^  sin^=ft)2  sin</)  — G)jcos0, 
^^  +  ^cos^  =  a,3, 


(4) 

(5) 
(6) 


and  it  is  known  that  A=B,  since  the  top  is  symmetrical  about 
its  axis  ;  and  that  Wg  has  an  initial  value  ;/  given  to  it  in  spin- 
ning, while  6  has  an  initial  value  0q,  at  which  inclination  to  the 
vertical  we  place  the  top  at  the  beginning  of  the  motion. 


1.  i 


(I) 

(2) 

(3) 


(4) 

(5) 
(6) 


MOTION   ABOUT   A   FIXED   POINT.  115 

Equation  (3)  becomes 

6  — "*=o. 
dt 

.'.  61)3  =  constant  =  //,  its  initial  value 

Equations  (i)  and  (2)  give,  when  multiplied  by  to^,  to^,  respec- 
tively, and  added  together, 

AI  <>>i-Tf  +  c^a ^f )  =  ^^S^''  ^^^  ^(*^i  sin  </> 4- <»2  ^^^  ^)» 
which,  by  aid  of  (4),  becomes 

This,  on  integration,  gives 

A  \     2  w^rtfcoj -{-A  \     2  (o^^dw^  =  2  Mgh  I    sin  ^^^. 

.-.  Ai(o^-\-w^)  =  2Mgh{co^0Q-cos6). 
But,  taking  (4)  and  (5),  and  squaring  and  adding,  we  get 

. •.  a(^J  +  A  sin2  df^y  =  2  3fgh{cos  Oq-cos  6), 


(a) 


which,  as  will  be  seen  hereafter,  is  one  form  of  the  equation  of 
energy. 

6^.    In   order  to  obtain  another  relation   between  -,  ,    —f' 

^  dt      dt 

we  may  proceed  as  follows. 

Multiply  (i)  by  cos^  and  (2)  by  sine/)  and  subtract,  and  we 

get 

,  dw„  .  do),     C—  A    dd  . 

sin</.^-cos0-J  =  -^;.-^-.  from  (4). 


m 


^1 


n  f 


W: 


Ii6 


But  by  (5), 


lit 


RIGID   DYNAMICS. 


sin  0  =  &)2.sin  <^  — Wj  cos  <^. 


and  since  by  (6) 

—r  =  u y-  cos  ff, 

dt  dt 

dyjr        ^dO       .     r^d^ylr     Cn     dO 
dt  dt  dt^      A      dt 

and  therefore,  multiplying  through  by  sin  6  and  integrating,  we 
have 


A  sm^d~y^=Cn{cos  ^j,  — cos  6), 


{b) 


since,  initially,  the  value  of  the  left-hand  side  of  the  relation 
is  zero. 

66.  The  relation  {b),  which  may  also  be  obtained  geometri- 
cally, shows   that    the   sign    of   ^  depends  upon  the  sign  of 

;/(cos^Q  — cos^),  since  ^  and  C  are  positive  quantities.  In  the 
case  we  have  supposed,  cos  6^  is  always  greater  than  cos  6, 
since  6^  is  the  least  angle  the  axis  of  the  top  can  nake  with 
the  vertical ;  if  the  top  were  spun  so  that  the  centre  of  gravity 
were  below  the  fixed  point,  then  cos^^<cos^.     Thus  we  see 

that  the  signs  of  -^  and  of  twg  will  be  the  same  or  opposite, 

according  as  the  centre  of  gravity  is  above  or  below  the  fixed 
point.  This  is  equivalent  to  saying  that  tlic  motion  of  pre- 
cession which  the  top  acquires  is  direct  or  retrograde  according 
as  the  centre  of  gravity  is  above  or  below  the  fixed  apex  about 
which  it  moves. 

This  motion  of  precession  and  its  sign  can  easily  be  shown 
by  a  top  of  special  construction,  which  is  so  arranged  that  one 
can  alter  at  will  the  position  of  its  centre  of  gravity. 


MOTION    ABOUT    A    I'lXED    I'OINT. 


117 


A  section  of  the  top  is  shown  in  Fly;.  51.  It  consists  of  an 
axis  of  steel  Ali,  pointed  at  A  and  B,  to  which  is  attached 
a  thick  conical  shell  S,  of  brass,  with  flanges;  a  sliding;  weij^ht 
C  can  be  moved  alonj:;  the  axis.  Without  the  slider  C,  the 
centre  of  gravity  of  the  top  is  nearly  at  the  point  B,  and  thus 


Fig.  51. 


by  moving  C  up  and  down,  it  can  be  made  to  fall  either  above 
or  below  the  point  B,  or  to  coincide  with  it. 

The  top  is  spun  by  holding  it  in  the  position  shown  in  the 
figure  between  an  arm  ADE  (movable  about  a  hinge  at  E) 
and  a  fixed  upright  with  a  small  cup,  roughened  on  the  inside, 
in  which  the  point  B  rests. 

A  string  is  wound  about  the  axis,  and  the  arm  ADE  being 
held  lightly  in  position,  the  top  is  spun  by  pulling  the  string; 
and  the  arm  being  then  removed,  it  remains  spinning  about 
the  point  B  and  exhibits  the  motions  of  precession  indicated 
b)  the  theory. 


I  1 


ii8 

RIGID   1 

)YNAMICS. 

It  may  be  noticed  here  that 

if  the  centre  of  gravity  be 

exactly 

at  the 

point  /?, 

and 

the  top  be 

accurately 

made,  its  axis 

will  be- 

come 

a  pcruiancnt 

axis,  and 

no  motion 

of  precession 

will  be 

seen, 

the  top 

while   spinning 

preserving 

the  position 

initially 

given 

to  it. 

t\ 


ii 


67.  The  motion  of  the  top  after  it  has  been  set  spinning 
and  placed  on  the  plane,  may  be  completely  determined  ex- 
plicitly from  the  initial  conditions  and  the  two  relations  {a)  and 
(jb)  just  obtained  : 


'<!h^^--<^> 


^{^^=2Mgh{zo^  ^^-cos  0), 


—  ,  ^^,  and  then  the  position  and  the  motion  of  the 
(//     dt 

top  at  evf^ry  instant  are  knov 


A  sin2 6/'^  =  0/(cos  <9o-cos  0). 
lit  ^ 

Ihese  give    —,  — f- 
d/     dt 

top  at  ev^ry  instant  are  known. 

As  we  have  seen,  — -^  depends  for  its  sign  on  ;/  and  the  posi 

dt 

tlon  of  the  centre  of  gravity;  it  also  changes  with  Q\  and,  or 

eliminating    -^,  we  get 
dt 

A  sin  O-j-  =  Vcos  Q^  —  cos  6  y/2Mgh'Am\^6 

,  de    . 

and  -—  wi 


.rj 


-  C'^;i^{cos  6q  —  cos  6) ; 

JQ 

and  —  will  also  change  in  value,  and  will  have  minimum  values 
dt 

(o)  when  0  =  0^^  and  6  =  dp  $1  being  a  root  of  the  quadratic 
2  A  Mgh  sin^  0  —  C'^ifi  (cos  9^  —  cos  Q)  =  o. 

The  top  will  then,  as  it  is  first    placed  on  the   plane,  tend 

dQ 
to   drop   down,  and    —    will   go   on    increasing    until,    having 

dt 

passed    some   maximum  value,  it   reaches   its   minimum  value 


MOTION    AHOirr   A    FIXKO   POINT 


119 


J 


when  B  =  Oy     Meanwhile  '  j   has  also  been  going  through  peri- 
odic changes,  being  a  maximum  when  6=$^. 

The  top  then  oscillates  between  the  positions  $^  and  6^,  and 
at  the  same  time  is  carried  about  a  vertical  axis  with  a  pre- 

cessional  motion   (not  constant)  '-^^ 

'   lit 
To  an  observer  placed  above  the  top  and  watching  the  pro- 
jection of  its  centre  of   gravity  on  the  horizontal    plane,  that 
point    would    describe    the   curve    indicated    in    Fig.    52,   lying 
between  two  circles  whose  radii  are  // sin  ^^  and  // sin  ^j. 


Fig.  52. 


The  curve  described  will  not  necessarily  be  closed ;  that 
will  depend  on  «  being  an  integral  part  of  2  tt.  It  is  evident 
also,  from  the  fact  that  maximum  and  minimum  values  exist  at 


I20 


KKJII)    DYNAMICS. 


the  cusps  and  the  outer  points,  that  the  curve  described  touches 
one  circle  and  cuts  the  other  at  ri;;ht  angles. 

The  maximum  value  of     j^  may  be  found  by  putting    ,  =o 
in  the  equations  on  page  1 18,  which  will  give 


II.. 


d  s i n*"^ e f "'^ Y  =  2  Mgh (cos  e^ -cos  6), 


A  sin^  B   J  =  Cn{cos  ^,,  —  cos  0). 


i<  I 


(it        Cn  Cn 

^F  being  the  weight  of  the  top. 


dO 


dyjr 


When  6  =  0,.,  it  can  be  seen  that  both   —  and  — "^    vanish 
identically. 


68.    Top  spinning  ivit/i  Great  Velocity  on  a  Rough  Horizontal 
Plane. 

In  most  cases  the  top  is  spun  with  a  very  great  velocity,  and 

then  placed  on  the  plane.  By  taking  the  value  of  -  already 
found, 

A  sin  e~  =  Vcol^l'T^^co^  V3  A/gliA  s'^u'^0-  CV(cos  ^^-cos  6), 

it  will  be  seen  that  if  ;/  become  very  great,  cos  ^q  — cos  ^  must 
become  very  small  in  order  that  the  expression  under  the  radi- 
cal may  remain  positive,  hence  the  axis  of  the  top,  instead  of  per- 
forming large  oscillations,  will  depart  but  little  from  0^,  its  initial 

position,  and  -^  will  approach  a  constant  value,  and  the  motion 
at 

will  therefore  become  steady.  The  time  of  a  small  oscillation 
may  be  found  in  the  following  way : 


til  ' 


=  0 


MOTION    AlUH'T   A    FIXKI)    POINT. 
Let  0  =  0^  +  //^  ,f  being  small. 


121 


COS  0^  —  COS  0 


im0 


=  u 


approximately,  and  the  foregoing  relation  for  '^^  becomes 


A<^  __-^  /t"S  0^  —  C(  )S  0 


dt 


'i^ 


MgliA  sin  0~  ChP''^^^  ^o-cos  0 

sin  0 


where 
Hut 


0 

sin  0 

,     ^({0        —  ^ 

•  "^  Jr^'^  '^Ig'tA  sin  0^11  -  ChihiK 

.A     d0_    . 

^_MghA9:m0, 

d0  _  dn 
dt  ~  dt 

die 


A 


=  at. 


^'^     V2  an  —  ir 


H  =  a  vers  — /", 
A 


and 


<— f') 


Q=0Q-\-a\\  —  cos^^/ 


This  is  a  periodic  function  which  repeats  values  of  6  every 
time  /  is  increased  by 

2  7r 

A 

and  therefore  the  time  of  a  complete  small  oscillation  is 

277  A 

Cn 


122 


Also, 


RIGID   DYNAMICS. 


dyjr  _  Cn     cos  0^^  —  cos  6  _  Cn 


dt      A 


•An^e 


Cn 


dt     A  sin  6. 


Chi' 


ii 


yi      sin  ^n 


.       .     .ai  I— cos-f'/ 
A  sni  ^„    V  A 


dyjr^      Cn       _  MghA?:\x\0,,f 


Cn  . 

•cos-—  t 

A 


fcj  I 


\\  i 


)  ,1 


I  =  I 


JVfgh 
Cn 


I  —cos 


Cn 

A 


,      Mirk  ^    ]\T8:h  .   .    Cn  ^ 


and  consist  of  two  terms,  one  increasing  uniformly   with  the 
time,  the  other  very  small,  and  a  periodic  function  of  the  time. 
If  n  be  extremely  large,  we  have,  approximately, 

,     Mgh  . 

and  the  precession  is  then  nearly  constant  and  equal  to 

Wh 


Cn 


W  being  the  weight  of  the  top. 


69.    If,   then,   a  top  be  spun  with  very  great  velocity  and 
placed  on  a  rough  horizontal  plane,  inclined  at  an  angle  to  the 

vertical,  it  will  make  small  oscillations  in  time  — - — ,  and  at  the 

Cn 

same  time  will  revolve  about  a  vertical  axis  with  an  angular 


velocity  very  nearly  equal  to 


Wh 
Cn 


In  the  ordinary  case,  the 


oscillations  will  be  so  rapid  at  first  as  to  be  barely  visible  to  the 
eye ;  is  the  speed  diminishes,  owing  to  resistance  of  air  and 
friction  at  the  apex,  they  become  more  noticeable  ;  until  finally, 


MOTION   ABOUT   A    FIXED   POINT. 


123 


when  the  top  is  ^^ dying''  n  becomes  comparable  with  the  other 
quantities,  the  oscillations  become  wider,  and  the  formulas  of 
Art.  6"]  apply. 


70.    Top  spinning  on  a  Smooth  Hori-zonial  Plane. 

Let  a  top  {Fig.  53)  be  spun  and  placed  in  any  manner  on 
a  smooth  horizontal  plane,  and  let  its  position  after  any  time 


Fig.  53. 


t  has  elapsed  be  that  shown  in  the  figure.  It  is  acted  upon 
only  by  the  reaction  R  of  the  plane  and  its  weight  Mg  acting 
at  G,  the  centre  of  gravity ;  and  if  ^,  77,  ^  be  the  coordinates 
of  G,  the  equations  of  motion  of  translation  are 


Im'^^M'^- 


dt"^         dfi 


O, 


dt^  dt^ 


m 


li 


I'  ■< 


B 


124 


RIGID   DYNAMICS. 


d^ 


From  these  it  is  seen   that  — ^  =  constant  =  initial  value ; 

dt 

-^  =  constant  =  initial  value ;  and  if  therefore  any  horizontal 
dt 

motion  be  imparted    initially  to  the  centre  of  gravity,  it  will 

preserve  that  velocity  at  every  instant  thereafter. 

And,  since  ^=//cos^,  CG  being  equal  to  h,  and  d  being  the 

inclination  of    the  axis  of    the  top  to  the  vertical,  the  third 

relation  becomes 


^./V^cos^)^ 
dt^ 


g' 


...  /e=,/|^.+f^(i|?i^|. 


If!  ,i 


The  equations   of   motion    of   the  top  about  the   centre  of 
gravity  considered  as  a  fixed  point  are 


(i)  ^^  +  (C-^)&)2«3  =  ^'^sin<9sin<f), 
dt 

(2)  A^^{A-C)ui^w^  =  Rh^mezo%i^, 

(3)  a)3  =  «; 


and  we  have  also  the  relations 


de 


(4)  —-  =  cousin  ^4- <W2^^s^, 
dt 

(5)  -^sin^  =  (U2  sin  <^  — G)iC0S(^, 

d'i' 


of 


MOTION   ABOUT   A   FIXED   POINT. 


125 


Thus  it  will  be  seen  that,  considering  the  centre  of  gravity 
as  a  fixed  point,  these  equations  are  similar  to  those  previously 
obtained  in  the  case  of  a  top  spinning  on  a  rough  plane  ;  the 
only  difference  being  that  for  Mg  in  those  relations  we  have 
R  in  these. 

The  solution  is  therefore  similar  to  that  given  in  Arts.  64 
and  65. 

We  have  ;?  =  j/^+ j,/^(!(l£^) 


df'' 


M 


•^  ^  — /;  sm  ^^— —  A'cos  ^  —     •• 


lU^ 


\dt)  \ 


And  multiplying  (i)  by  wj  and  (2)  by  Wg-  ^^^  have 

^eoj  ^''^  +  ^«.,  ^  =  ^/^  sin  ^ { wj  sin  </) + 0)2  cos  0 1 
dt  "  dt 

dt 

.'.  A (q)  2  +  ft,;-i)  =  2  (r/i  sin  e  '^f^  dt 
*^  dt 

=  2(\Mgh  sin  e-Am  sin2  6>^  -J/^2  sin  ^cos  ofjj^.de. 
. :  A  (coi^  +  0)32)  =  2  Mg/i{cos  6^  -  cos  6)  -  M/fi  sin^  0  {^^. 

. ,  A  W  +  /^  sin2  e('^  +  Mh^  sin2  9 ("^j 
\dtj  \dt  J  \di 


Jt) 


—  2  Mgh (cos  Q^  —  cos  0) , 
and  the  other  relation  will  be  as  before  : 

^  sin2  ^ -"^  =  0/(cos  ^^  -  cos  6?). 
dt 

These  two  relations  give  the  solution  of  the  problem. 


126 


RIGID    DYNAMICS. 


M 


And  it  is  evident  that,  independent  of  its  motion  of  translation 

in  a  horizontal  direction,  the  centre  of  gravity  can  only  move  up 

and  down  with  an  oscillatory  motion  while  the  apex  describes 

on  the  plane  the  fluted  curve  already  obtained  in  the  case  of  a 

rough  plane  (Fig.  52),  the  values  of  Oq  and  6^  being  as  before 

j0 
those  which  make  —  a  minimum. 

^/ 

If  0)3  =  n  be  very  great,  the  discussion  is  the  same  as  before, 

and  it  can  easily  be  seen  that  the  apex  of  <"he  top  will  describe 

a  simple  circle  (approximately)  on  the  plane,  and  the  motion 

will  be  steady,  the  time  of  a  small  oscillation  and  the  period  of 

precession  being  obtained  as  formerly. 


\y 


hi 


1  •!' 


I 


y      I 

■  ! 
I 


li: 


71.  All  the  previous  results  obtained  theoretically  in  the 
case  of  motion  of  a  top  on  a  smooth  or  rough  plane  can  be 
verified  experimentally  by  having  a  number  of  tops  made  similar 
to  that  shown  in  Fig.  54. 


Fig.   54. 

A  circular  plate  of  brass,  a  quarter  of  an  inch  in  thickness, 
and  from  three  to  five  inches  in  diameter,  has  a  steel  axis 
through  the  centre.  The  centre  of  gravity  of  the  top  may  be 
from  one  to  two  inches  from  the  apex  on  which  it  spins,  and 
the  point  may  have  varying  degrees  of  sharpness. 

Everything  should  be  symmetrical  and  made  true,  so  that 


MOTION    ABOUT    A    FIXED    POINT. 


127 


The  top  is  most  readily  set  spinning  by  using  a  two-pronged 
handle  with  openings  through  which  the  axis  may  pass  :  a  cord 
put  through  a  hole  in  the  axis  and  wound  about  it,  is  pulled 
rapidly,  and  the  top  drops  with  a  high  speed  from  the  handle. 
A  little  practice  enables  one  to  spin  the  top  and  let  it  drop  on  a 
smooth  or  rough  surface  at  any  required  inclination. 

The  following  problem  may  also  be  examined  by  using  several 
of  these  tops  of  various  sizes,  and  with  points  of  varying  degrees 
of  sharpness : 

A  common  top,  zvJicn  spun  and  placed  oji  a  rougli  horizontal 
plane,  at  an  angle  to  the  vertical,  gradually  assumes  an  nprigJit 
position.     Explain  this. 

This  is  the  case  of  the  ordinary  peg  top  of  the  schoolboy, 
which  is  usually  made  of  a  cone  of  wood  through  which  passes 
a  steel  axis  ending  in  a  sharp  point  ;  when  spun  upon  a  rather 
rough  surface,  it  gradually  becomes  upright  and  'sleeps.' 

It  will  be  found,  after  a  little  experimenting,  that  this  appar- 
ently paradoxical  rising  of  the  top  to  a  vertical  position  against 
the  force  of  gravity  depends  on  two  things  : 

1.  The  degree  of  sharpness  of  the  apex  on  ivhich  the  top  spins. 

2.  The  position  of  the  centre  of  gravity. 

If  the  point  be  very  sharp  so  that  the  top  in  spinning  is  not 
able  to  form  a  small  conical  bed  for  itself  and  thereby  be  acted 
on  by  a  couple  arising  from  friction  at  a  considerable  distance 
from  the  point,  it  cannot  possibly  become  erect. 

When,  however,  the  point  is  rather  blunt,  and  the  centre  of 
gravity  not  too  high,  the  top  will  slowly  rise  up  under  the  action 
of  the  friction  (which  tends  to  diminish  the  angle  of  inclina- 
tion), and  'sleep.' 

The  equations  of  motion  are  similar  to  those  obtained  in  Art. 
64,  with  the  additional  relations  introduced  by  friction. 

The  solution  of  the  equations  shows  that  the  top  rises  to  the 
vertical,  on  the  supposition  that  the  point  of  the  top  is  a  portion 


■"  ■■  ■  1  :'■ 

"M 


;  s 

t,  . 


!|-i>' 


128 


RIGID    DYNAMICS. 


.il 


r  i 


of  a  spherical  surface  and  that  friction  is  thus  enabled  to  act  in 
the  proper  manner. 

A  complete  analytical  solution  of  the  problem  is  given  in 
Tellett's  T/u'ory  of  Friction,  Chrp.  VITI.,  where  the  top  is  sup- 
posed to  be  a  symmetrical  pear-shaped  cone  with  a  spherical 
surface  as  the  apex  upon  which  it  spins. 


Fig.  5£ 


•HI 


72.  The  Gyroscope  nwving  in  a  Horizontal  Plane  about  a 
Fixed  Point. 

If  a  gyroscope  bfe  put  in  rapid  motion  and  placed  so  that  the 
prolongation  of  the  axis  of  rotation  can  rest  on  a  fixed  point  of 
support,  and    if,  at  the  same  time,  an  initial  angular  velocity 


MOTION   ABOUT   A   FIXED    POINT. 


129 


;t  in 


a  in 
sup- 
rical 


about  the  point  of  support  be  given  bodily  to  the  gyroscope  (in 
the  proper  direction)  in  a  horizontal  plane,  it  will  revolve  about 
a  vertical  axis,  and  the  apparently  paradoxical  motion  is  pre- 
sented of  a  body  whose  centre  of  gravity  moves  in  a  horizontal 
plane  although  its  point  of  support  is  at  quite  a  distance  from 
the  vertical  through  the  centre  of  gravity. 

In  Fig.  55  the  gyroscope  is  supposed  to  be  set  rotating  and 
started  in  a  horizontal  plane  with  its  centre  of  gravity  at  the 
point  G,  the  weight  acting  vertically  downwards  in  the  direc- 
tion indicated  by  the  arrow.  It  is  supported  only  at  the  point 
O,  and,  if  rotating  rapidly  enough,  will  keep  on  moving  uniformly 
in  this  horizontal  plane  in  a  direction  hereafter  determined. 

Its  position  at  any  time  is  given  by  the  position  of  its  />riu- 
cipal  axes  at  O :  these  are  OA,  OB,  OC. 

It   is  evident  that  ^  =  —  and  that   C  moves  along  XNN\ 

2 

NON'  being  the  line  of  nodes,  and  the  angle  BON  =  </>. 

At  each  instant  the  gyroscope  tends  bodily  to  turn  about 
NON'  under  the  action  of  gravity,  and  the  value  of  this  turning 
couple  is  mgh,  tn  being  the  mass  of  the  gyroscope  and  OG  =  h. 

Resolving  this  couple  mgli  into  two,  we  get 


;  \, 


i    ! 


mgh  cos  ^  about  OBy 
and  7ngh  sin  ^  about  OA. 

Then  Euler's  equations  become  : 


vit  a 

t  the 
nt  of 
ocity 


J 


A  — -  -f  {C—  /4) W2«3  =  nigh  sin  ^, 


A     2 


dt 


(C—  ^)a),a)3  =  tngh  cos  0, 


^  dt     °' 


from  which  it  is  seen  that 


a»3  =  constant  =  «. 


■ '  U' 


:ttJ 


V  ' 

l'  ''i 
fl'  -  i' 


130 

Also,  we  have 


RIGID   DYNAMICS. 


-  =0  =  <Uj  sin  ff)-\-a>,^  cos(j>, 
dt 

;  =&).,  sin  <p  — ft),  coso, 

dt     •  ^       ^       ^        ^ 

d(}> 


dt 


=  ;i, 


since  6—  -,  and  therefore  cos^=o. 


h 


it-  ■•  i 


m 


■f 


1    4 


■h 


i    I 

!     I 


From  the  preceding  relations  we  have : 
a)j  sin  (fy  +  o)^  cos  ^  =  o, 


ft). 


sin  <^  — ft)j  cos<^  = 


rA/r 


.-.  squaring  and  adding 


But  since 


2^     2     ^^W 
ft>f +  ftJ3^=    -  r 


^  -7-  +  {C—  A)(o^(o^  —  vtgh  sin  <^, 


A      ,^  —  {C—A)(i),COo  =  JUg'/lCOS(l). 

dt 

Therefore,  multiplying  the  former  by  (o^,  and  the  latter  by  q)^, 
and  adding,  we  get 

Aco^-^  +Aco^-~  =  7ng/i{co^  sin  (fi  +  w^  cos(f>)=0. 


/^  (&)j2  +  0)2^)  =  constant. 

.•.  G)j2 +  &)./  =  its  initial  value,  =  «^  say. 


Vi-  : 


f  I 


Then 


d-\lr  ,  d(b 

—y-  =  a,  and  — -  =  «. 
dt  dt 


MOTION   ABOUT   A   FIXED   POINT. 


131 


3y  «2' 


since  both  may  be  taken  zero  when  /  is  zero. 

. ■.    (o^—  —a  cos  «/, 
(Of^  =  a  sin  nt, 

And,   substituting  these   values   in   the   relation   for  the  first 
couple,  we  get 

A{an  sin  «/)  +  {C—  A)n[^n  sin  «/)  =  m^/i  sin  «/. 

. '.   Cnu  sin  ;//  =  w/^//  sin  « A 


and 


d^  _jngh  _  Wh 

~  dt  ~  Cn       Cn ' 


TF  being  the  weight  of  the  top,  and  ;/  being  the  initial  velocity 
of  rotation.     Hence  the  axis  OC  moves  around  in  a  horizontal 

plane  with  uniform  velocity  ---^  and  the  direction  of  revolution 

is  indicated  by  the  sign  of  71  or  Wg ;  that  is,  /o  an  observer 
looking  dozvn  in  the  direction  ZO,  the  gyroscope  ivill  revolve 
bodily  in  the  same  direction  as  the  gyroscope  rotates  about  its 
axis  luhen  viewed  by  an  observer  at  C. 

It  is  important  to  observe  that  the  necessary  condition  for 
the  motion  of  the  gyroscope  bodily  about  OZ  is  that  it  receives 
an  initial  angular  velocity,  so  that 


O) 


i+(>>'i-\--f-]  =some  finite  quantity. 


If  this  initial  velocity  be  not  given  to  it,  it  will  act  in  the 
same  way  as  a  top,  tending  to  drop  down  and  oscillate  as  it 
moves  around  the  vertical. 


132 


RIGID   DYNAMICS. 


ii 


^ 


I 


U'l  i< 


I'  ' 


Usually  w  is  very  great,  so  that  «  is  small,  and  the  preces- 
sional  motion  is  slow. 

I'^or  a  complete  discussion  of  the  experiments  which  can  be 
performed  with  the  Gyroscope  see  Chap.  X. 

73-  ^^  fi'^^^  ^/'^'  Pt'cssurc  on  the  Fixed  Point  in  the  Case  of  the 
Gyroscope. 

As  an  illustration  of  the  use  of  the  equations  of  Art.  63,  we 
may  find  the  pressure  on  the  point  about  which  the  gyroscope 
revolves. 

In  this  case  we  shall  have,  calling  the  mass  of  the  gyroscope 
5  to  avoid  confusion, 

=  P  cos  X  +  S/z/A'-  s{^z-^^^, 

S  \  CO,  .  C+A-b(^  +-^^)  -  (0,3'-=  +  a,,V I 
=  Pcos^  +  ^mV-s(^.v-^cy 

.  =  P  cos  v  +  'LmZ—S(—y -'^xj, 

which  become,  since  A  =  B,  and  (x,  y,  z)  are  (o,  o,  h), 

s\<o,-  C-^^\=P  zos\  +  Sgcos^-s\^^^'  h^\, 
5|a,2.C'/^}=Pcos/i+<>rsin<^  +  5|^"-^./.2|. 

S\—{(i)^-\-(t)^)h\=PC0'iiV, 

the  last  of  which  can  be  obtained  from  elementary  consider- 
ations. 


\-  ■  }: 


MOTION    AHOUT    A    FIXKI)    I'OINT. 


133 


D        N  (///;'■  COS  (i    ,,,  ,  ,       /^hn) 

'        //  '  A  ) 

/'  cos  /A  =  /;/    -    -^    ,— ^  •  /r  -.V  sin  0  +  w./         , 
'  A  A  ) 

.  P  cos  1/  = ;;/ 1  —  ((Uj-  +  w.^)/t  \ , 
the  mass  being  denoted  by  ni. 

These  relations  taken  in  conjunction  with 

(Uj  sin  (f)  +  0)2  cos  0  =  0, 


tocsin  </>  — ^1  cos  0=    ,"  =  — 
</0 


<// 


=  //, 


give,  on  squaring  and  adding,  the  value  of  P  in  terms  of  known 
quantities. 

Similar  equations  may  be  obtained  in  the  case  of  the  top 
spinning  on  a  smooth  horizontal  plane. 


ClIAPTl'.R   VII. 


tli 


'  I 


MOTION   ABOUT   A    FIXKH    F'OINT. 
Impulsive  Forces. 

74.    In  forminfi;  the  general  equations  of  motion  for  finite 
forces,  we  had  two  sets  of  relations  of  which  the  types  are 


and 


2.m      ..  =    -  2;;/  -'  =  2;//^'-f-  P  cos  \, 
(it-     lit         dt 

V     (    d'h       dh'\       d^     \    da       dv)      J. 
r   dt^        dfi )      dt        r  dt        dt  I 


and,  remembering  the  definition  of  an  impulse,  we  get  our 
impulsive  equations  from  these  by  integrating  with  respect  to  /, 
from  o  to  T,  some  small  value  of  the  time. 

That  is,  instead  of  a  continuous  change  we  have  an  abrupt 
change  of  velocity  and  of  moment  of  momenuim  taking  place 
during  an  exceedingly  small  time  t. 

Hence,  for  impulses  X,  V,  Z,  we  get  the  equations 

=  Md  .  (CO J  -  0)^)  -  My{(oJ  -  o),) 
=  SA"+Pcos\, 

with  two  similar  relations  for  Y  and  Z.  These  determine  the 
impulse  J\ 

134 


i 


MOTION    ABOUT   A   FIXED   I'UINT. 


135 


For  the  couples  wc  have 


the 


L^£_^^f 


-'-"i-^;^-^!J/ 


But 


=  (n^rny^  4-  -'^  —  a)^S;;/.t7  —  ui^mxz 

=  /l(Oj.  —  F(Oy  —  Eo)^, 

.'.  wc  get,  for  the  three  couples, 

A  («;  -  o),)  -  /^((u;  -  cOy)  -  EicoJ  ~  CO,)  =  L, 
■  B{coJ  -  (Oy)  -  D{(oJ  -  ft),)  -  F{a>J  -  CO,)  =  M, 

CicoJ  -  ft,,)  -  E{C0J  -  CO,)  -  D{C0J  -  COy)  =  jV, 

CO,,  ft)y,  0),  being  the  digular  velocities  about  axes  fixed  in  space 
at  time  /,  and  these  being  suddenly  changed  by  the  impulsive 
actions  to  coj,  coj ,  coj. 

75.    Taking    the    foregoing    expressions    for    the    impulsive' 
couples,   we   can    simplify   them    by   choosing   principal   axes, 
which  make  D,  E,  F  vanish  ;  if,  at  the  same  time,  the  body 
starts  from  rest,  oi„  co^,  ft),  are  zero,  and  the  equations  become 

Aco',=:L, 
Bco\  =  M, 
Cco\==N. 

The  equations  of  the  instantaneous  axis  are 

X  _  y  _  c 


or 


or 


«x 

«'r 

CO,, 

X 

A 

y 

M 
B 

z 
C 

Ax 

.By_ 

Cz 

L 

M 

N 

!>-.' 


Ik:  . 


V  ' 


136  RIGID   DYNAMICS. 

The  plane  of  the  impulsive  couple  is 

and  therefore  the  instantaneous  axis  (that  is,  the  line  about 
which  the  body  will  begin  to  rotate  under  the  action  of  the 
impulse)  is  the  line  conjugate  to  the  plane 

with  regard  to  the  ellipsoid 

The  equations  of  the  instantaneous  axis  are 

L      M     N' 
and  the  equations  of  the  axis  of  the  impulsive  couple  are 

X _  y  _z 

Hence  it  will  be  seen,  by  comparing  these  two  sets  of  rela- 
tions, that  if  a  body  fixed  at  a  point  be  struck,  it  will  not  begin 
to  rotate  about  the  axis  of  the  impulsive  couple  induced  by  the 
blow,  unless  A—B=C,  or  unless  tlie  plane  of  the  impulsive 
couple  be  a  principal  plane  or  parallel  to  a  principal  plane. 
For  the  two  sets  cannot  reduce  to  a  single  set  unless  A  =  B=C\ 
or  unless  two  of  the  quantities,  x,  y,  z,  vanish,  (which  means 
that  the  axis  of  the  couple  is  one  of  the  principal  axes). 

It  v/ill  be  seen  from  the  preceding  investigation  that,  if  a 
rigid  body  be  free  to  turn  about  a  fixed  point,  the  problem  of 
determining  the  change  produced  in  the  motion  of  the  body  by 
the  action  of  a  given  impulse,  is  equivalent  to  determining  the 
change  in  its  motion  when  the  body  is  acted  on  by  a  given 
impulsive  couple.  This  equivalen:e  also  appears  from  the  fol- 
lowing considerations.     The  impulse  may  be  resolved  into  an 


I 

\ 


i 


MOTION   ABOUT   A   FIXED   POINT. 


137 


* 


equal  and  parallel  impulse  acting  at  the  fixed  point  and  an 
impulsive  couple.  The  impulse  acting  at  the  fixed  point  will 
have  no  influence  on  the  motion  of  the  body,  and  therefore  only 
the  couple  need  be  considered.  Resolving  the  latter  with 
respect  to  the  coordinate  axes  we  obtain  the  equations  on 
page  135. 

Illustrative  Examples. 

I.  A  cube  is  fixed  at  its  centre  of  inertia,  and  struck  along 
an  edge. 

In  this  simple  case  it  is  evident,  without  forming  the  equa- 
tions of  motion,  that,  since  the  momental  ellipsoid  is  a  sphere, 
A  =  B=C,  and  the  cube  begins  to  rotate  about  the  axis  of  the 
impulsive  couple. 

Thus,  in  Fig.  56,  the  cube  is  fixed  at  O,  its  centre  of  inertia, 
and  on  being  struck  by  a  blow  Q,  begins  to  rotate  about  the 
axis  of  the  impulsive  couple  A  OB. 


\ 

N 

^ 

q/^ 

n 

\ 

\ 

Fig.  56. 

2.  A  homogeneous  solid  right  circular  cylinder  is  rotating 
with  given  angular  velocity  about  its  centre  of  inertia,  which  is 
fixed ;  the  cylinder  receives  a  blow  of  given  intensity  in  a  direc- 


ill 


m 


138 


RIGID   DYNAMICS. 


li:,' ' ' 


111 


tion  perpendicular  to  the  plane  in  which  its  axis  moves.     Deter- 
mine the  subsequent  motion. 

3.  A  lamina  in  the  form  of  a  semi-ellipse  bounded  by  the 
axis  minor  is  movable  about  the  centre  as  a  fixed  point,  and 
falls  from  the  position  in  which  its  plane  is  horizontal ;  deter- 
mine the  impulse  which  must  be  applied  at  the  centre  of  inertia, 
when  the  lamina  is  vertical,  in  order  to  reduce  it  to  rest. 

If  this  impulse  be  applied  perpendicularly  to  the  lamina,  at 
the  extremity  of  an  ordinate,  through  the  centre  of  inertia, 
instead  of  being  applied  at  the  centre  of  inertia  itself,  show 
that  the  lamina  will  begin  to  revolve  about  the  major  axis. 

4.  A  triangular  plate  (right  angled)  fixed  at  its  centre  of 
inertia  and  struck  at  the  right  angle  perpendicularly  to  the  plate. 


u  i 


Fig.  57. 


In  Fig.  57  let  G  be  the  centre  of  inertia  of  the  trangle,  and 
C  the  point  where  the  blow  is  struck  at  right  angles  to  the  plane 


MOTION   ABOUT   A   P^IXED   POINT. 


139 


of  the  paper.  Then  if  we  construct  the  momental  ellipse  at  G, 
it  touches  the  three  sides  at  their  middle  points.  The  impulsive 
couple  in  this  case  contains  the  line  CG  in  its  plane ;  but  since 
AB  is  a  tangent  to  the  ellipse,  A'GB'  is  the  diametral  line  con- 
jugate to  CG.  The  triangle  therefore  commences  to  rotate 
about  A'GB',  which  is  drawn  parallel  to  the  hypothenuse. 

5.  A  solid  ellipsoid  fixed  at  its  centre  is  struck  normally  at  a 
point/,  g,  r. 

If  /,  m,  n,  be  the  direction  cosines  of  the  line  of  the  blow 
whose  magnitude  is  Q,  and  if  the  equation  of  the  ellipsoid  be 

/V'M  Mill  rvj 

a^     Ir     c^ 

then  the  equations  of  the  instantaneous  axis  will  be 

Ax^By_^Cz 
L      M     N' 

or  ^^{l^  +  c\r 

Q{qn  —  rvi)  ' 

qn  —  nn       rl  —pn       p  in  —  ql ' 

and  since  the  blow  is  normal  to  the  ellipsoid  at  /,  q,  r, 

I      VI      n 
du     d?i     dii 
dx     dy     dz 


or 


/  _ ;«  _  n 

p      q     r 


a" 


^2 


Therefore  the  equations  of  the  instantaneous  axis  will  be 
^     U^^c^    ^q_     c^  +  d^  ^^r_     a^^b'^ 

^2  •  ^2_,.2'^  ^2  •  ^_^iy  ^2  • 


a^-lP- 


ii 


i:i 

t: 


! 

'I 

t 


CHAPTER  VIII. 

MOTION   ABOUT   A  FIXED  POINT.     NO   FORCES  ACTING. 

76.    Heavy  Body  fixed  at  its  Centre  of  Gravity. 

The  simplest  case  of  motion  under  no  forces  which  ordinarily 
presents  itself  is  that  of  a  body  acted  on  by  gravity  and  fixed 
in  such  a  manner  that  it  can  only  rotate  about  its  centre  of 
gravity  considered  as  a  fixed  point. 

Here  we  have 

at 

And,  multiplying  these  three  equations  by  eDj,  ©g,  6)3,  respec- 
tively, and  adding,  we  get 

dt  dt  dt 


.'.  Aw^+B(o^-\-C(o^=  a  constant 


(I) 


Similarly,  multiplying  the  three  equations  by  A^i,  Bm,^.,  Cco^, 
respectively,  adding,  and  integrating,  we  get 


A^a)^-hB^o)^^-\-C^(o^^=  a  constant 
140 


(2) 


'3' 


(2) 


MOTION   ABOUT   A   FIXED   POINT. 


141 


(i)  States  that  the  kinetic  energy  is   constant,  as  might  be 
expected,  since  no  forces  act ;  this  can  be  seen  by  taking 


I"  Smir 


■-^MO'-m^m\ 


=  |-Swf(ra.2— J<»3)^+  •••  +  •••  }2 

since  the  products  of  inertia  vanish. 

(2)    is   another  way   of    expressing    the    constancy  of    the 
moment  of  momentum. 

For  (moment  of  momentum)^ 


where 


0)1 


'2' 


77.  Now,  since  h^,  h^,  h^  are  constant  at  all  times,  the  plane 
h^x-^h^y-\-h^z-o,  or  A(ii^x-\-B(d^y^Cai^z=o  is  an  Invariable 
Plane  fixed  in  space ;  the  line 

X  _  y 


A( 


Bo)o    Ci 


Wc 


iO)i       x^t«2       '-"'3 

is  perpendicular  to  this  plane,  and  is  an  Invariable  Axis. 
The  instantaneous  axis  is  given  by 


X 


6)1 


0)2      0)3      0) 


i 
till 


I  .:, 


'*     ! 


(       I 


ill 


I'  '  ' 


Ir't'' 


HIT 


142 


RIGID   DYNAMICS. 


78.    If  we  now  construct  the  momental  ellipsoid  at  the  fixed 
point  O,  as  in  Fig.  58,  O  '',  OB,  OC  being  the  principal  axes, 


Fig.  58. 


and  POP'  the  instantaneous  axis  at  any  time  /,  the  equation 
of  the  ellipsoid  will  be 

and  those  of  the  instantaneous  axis 

ft)j     Wg     a>3     ft) 

Now,  X,  y,  3  being  any  point  on  this  line,  let  it  represent 
the  point  P ;  then  at  P  we  have 


MOTION   ABOUT   A   FIXED   POINT. 


143 


i-  —  Ji — _£.  — '' 

Wj  ft)^  (Ug  ft) 


and 


.'.    G)  =  _  •  r, 


,r  =  G) 


^■/&' 


J  =  «..^, 


Ml 


r=«Wo 


k 


Therefore  the  angular  velocity  at  any  instant  is  proportional 
to  the  radius  vector  of  the  ellipsoid. 

Moreover,  taking  the  tangent  plane  at  P  to  the  ellipsoid,  its 
equation  is 

ax  dy  dz 


where 

which  becomes 


^=«i|,  y^^i\'  ^^^^j' 


^-"4)l+-+-=°' 


or 


or 


Ao).^  ■  ^  +  B(o^  .  rj  +  Cco^  •  ^=kc. 
And,  if  we  construct  the  plane 

Aa^x + Bw^y  +  C(o^s  =  o 


and  represent  It  by  XYX'Y',  this  is  the  invariable  plane ;  and 
we  see  that  the  tangent  plane  to  the  momental  ellipsoid  at  the 


1^1 


il 


ItiH 


;'';' 


If! 


144 


RIGID   DYNAMICS. 


point  where  the  instantaneous  axis  cuts  the  ellipsoid  is  always 
parallel  to  this  invariable  plane. 

Hence,  the  motion  of  the  body  fixed  at  O,  and  under  the 
action  of  no  forces,  is  completely  represented  dy  tJiv  rolling  of 
the  momcntal  ellipsoid  on  a  plane  fixed  in  space  and  parallel  to 
the  invariable  plane,  and  at  a  distance  from  it  equal  to  00\ 


sir  ^ , 


If..' 


J'-l: 

II'. 

[I  »■ 


',1  I' 


I ' 


79.  The  ellipsoid  in  rolling  on  the  fixed  plane  traces  out  a 
curve  on  that  plane,  and  also  one  on  its  own  surface. 

The  curve  traced  out  on  the  surface  of  the  ellipsoid  is  called 
the  Polhode,  and  its  equation  is  found  by  taking  the  condition 
that  the  perpendicular  from  the  centre  of  the  ellipsoid  on  a 
tangent  plane  at  x,  y,  a  is  constant,  and  combining  it  with  the 
equation  of  the  ellipsoid  itself. 

The  equation  of  the  Polhode  is,  therefore, 

Ax'^+By'^+Cz'^^c^ 
A\v'^  +  E^y'^^-(?d^=c'\ 

The  curve  traced  out  on  the  plane  is  called  the  Herpolhode^ 
and  its  equation  is  found  from  the  relation 

a  pi = p2  =  Opi  -00'^  =  t^  -p\ 

and  will  vary  with  r,  and  therefore  with  ta  and  with  p. 

It  is  apparent  that  any  one  of  the  central  ellipsoids  might  be 
chosen  instead  of  the  momental  ellipsoid,  and  the  motion  of  the 
body  exhibited  in  a  similar  manner  by  the  changes  in  motion  of 
the  ellipsoid  chosen. 

Innumerable  problems  may  be  constructed  from  the  preceding 
representation  ;  but  they  are  all  dependent  on  properties  of  the 
ellipsoid,  and  are  not  problems  in  Dynamics. 


CHAPTER   IX. 


MOTION   OF   A   FREE   BODY. 


80.    We  have  already  seen,  in  discussing  DAlcmbcrfs  Prin- 
ciple, that  the  general  equations  of  motion  of  any  body  are 


M^-^=- 


X 


2;;^(  Z 


and 


-;« 


w- 


S;«{.( 


X 


dfl 
d\x 


-dt-n^ 


y  _d^y 
dt"^ 


^m[x{  Y 


dfi 
d'^y 


—  X 


Z 


dfi 


-Ax 


dty\ 
dh 


=0, 


=0, 


d\x\ ) 


If  M  be  the  whole  mass,  x,  y,  ^  the  coordinates  of  the  centre 
of  inertia  at  time  t,  and  x',  y\  2'  the  place  of  m  relatively  to  a 
system  of  axes  originating  at  the  centre  of  inertia  and  parallel 
to  the  original  set  of  axes,  then  the  equations  of  motion  become 

d^ 


H5 


if 


\  \ 


«ll 


i : 


!-!    I 


146 

and 


RIGID   DYNAMICS. 


2;;/ 


2;//!,y[.r 


..{.(K-'|^)-y(.v--)}=o, 


which  latter  can  be  transformed  in  the  ordinary  way  so  as  to 
determine  the  angular  velocities. 

These  equations  theoretically  give  a  complete  solution  of  the 
problem. 

But  the  most  important  case  of  free  motion  of  a  body,  and 
the  only  one  which  admits  of  simple  solution,  is  that  in  which 


o 


Fig.  59. 


■t/de  particles  of  the  body  7nove  iji  parallel  planes.  Here  it  is  evi- 
dent that  we  need  only  consider  the  motion  of  one  particular 
plane  of  particles,  and  that  containing  the  centre  of  inertia  is 


as  to 

f  the 

,  and 
/hich 


.^ 


>  evi- 
:ular 
ia  is 


MOTION   OF   A   FKEK    I50DY. 


H7 


chosen,  and  the  position  of  the  body  at  any  time  determined  in 
the  following  way. 

Let  the  plane  in  which  the  centre  of  inertia  moves  be  repre- 
sented by  the  plane  of  the  paper,  the  same  section  of  the  body 
being  represented  at  any  two  times  as  in  Fig.  59. 

Let  the  body  be  referred  to  fixed  axes  OX,  OV,  and  let  AC/> 
be  any  line  in  the  body  passing  through  the  centre  of  inertia 
C,  and  in  its  initial  position  let  this  line  be  parallel  to  OV,  as 
shown.  Then,  after  any  time  f,  the  body  has  reached  its  second 
position,  and  it  is  evident  from  elementary  geometry  that  the 
body  can  get  from  its  first  position  to  the  second  by  translation 
of  the  centre  of  inertia  C,  and  by  rotation  about  C  through  an 
anjrle  0,  equal  to  that  which  ACB  in  Its  second  position  makes 
with  the  axis  O  V,  or  with  a  parallel  line  fixed  in  space. 

For  translation  of  the  centre  of  inertia,  we  have,  by  D'Alem- 
bert's  principle, 


dt'' 


dt'' 


2.1)1 — -^  =  Zm  V  =  AT — 4- 
dt^  d*^ 

And  for  rotation  about  the  centre  of   inertia  considered  as  a 
fixed  point,  we  get 


i  dt"   ^ dt^S  dt^ 


\ 


X 


Therefore,  at  any  time,  the  motion  of  the  body  will   be  fulIyjQ 
known  when  we  know 

1.  The  initial  conditions,  so  that  6  is  known. 

2.  The  coordinates  of  the  centre  of  inertia  with  reference  toi*  "I 


ar 


some  axes  fixed  in  space  ;  this  gives  — '--,    —^. 

dt^     dfi 

3.  Mk"^  about   the  axis   of    rotation   through   the   centre  of 

inertia. 

4.  Geometrical  relations  between  x,  y,  0 


i 


iK: 


I 


f'  '  I 


r'' 


II  ' 


•';!!! 


148 


RIGID   DYNAMICS. 


In  cases  of  constraint  where  bodies  roll  or  slide  on  others, 
geometrical  relations  are  easily  found,  and  the  unknown  reac- 
tions eliminated  by  taking  moments. 

Illustrative  Examples. 

I.  A  heavy  sphere  rolling  down  a  perfectly  rough  inclined 
plane. 

In  this  problem  gravity,  by  the  aid  of  friction  and  the  reac- 
tion of  the  plane,  produces  both  the  translation  of  the  centre  of 
inertia  and  the  rotation. 

Let  OXy  O  V  (Fig.  60)  be  the  axes  of  the  coordinates  fixed  in 
space,  the  sphere  starting  to  roll  from  O.     Then  at  any  time  /, 


Fig.  'jO. 

the,  position  of  the  sphere,  is  given  by.r,  j,  the  coordinates  of 
C,  the  centre  of  inertia,  and  the  angle  0  through  which  the 
sphere  has  rolled  ;  that  is,  through  which  it  has  rotated  about 
C,  considered  as  a  fixed  point. 

The  initial  conditions,  combined  with  the  geometrical  condi- 
tions for  perfect  rolling,  give 

x=ad,  y=a.  (i) 


MOTION   OF   A   FKEi:    BODY. 
For  the  translation  of  the  centre  of  inertia,  \vc  have 


rif-X 


at' 


M'-^ 


J  +  J4'-cosrt-A'  =  o. 


if' 


The  rotation  about  C  is  given  by 


149 

(2) 

(3) 


(4) 


These  four  relations  give  a  complete  solution  of  the  problem, 
for  we  have 


and,  therefore,  from  (2)  and  (4), 


M{k^^a^)'^^^=Arga^mii., 


(5) 


from  which  it  is  seen  that 


dfi 


=  f^sin  a. 


and 
also, 


:^=i\^sin  «•  /2; 
R  =  Mg  cos  a, 
F=|  i^i^sin  a. 


These  results  give  the  space  passed  over  in  time  t,  and  show 
that  five-sevenths  of  gravity  is  used  in  translation,  while  two- 
sevenths  is  used  in  turning  the  sphere  about  the  centre  of 
inertia. 

The  relation  (5)  may  also  be  obtained  at  once  by  forming  the 
equation  of  energy.  For  the  sphere  has  fallen  through  a  dis- 
tance X  sin  u.,  and  therefore  the  work  done  by  gravity  is 


[''  \ 


Ik 


ill ' 


l!i    ll    i 


I  It 


If      :  > 


150 


or 


RIGID   DYNAMICS. 
M^v  sin  a, 
MgaO  sin  «, 


which  must  be  equal  to  the  kinetic  energy  at  time  t,  and  there- 
fore 

1  MirJ^  +  /.■2a)2)  =  J/^rt^  sin  «. 

.-.  |i^/(.?H/f'2)C^Y  =  i]/^«(9sina, 
which  gives,  on  differentiation, 


M(a^^k')^^=Mga  sin  a. 


as  before. 


2.  If  a  heavy  circalar  cylinder  rolls  down  a  perfectly  rough 
inclined  plane,  one-third  of  gravity  is  used  in  turning  and  two- 
thirds  in  translation. 

3.  A  very  thin  spherical  shell  surrounds  a  sphere,  both 
being  perfectly  smooth  and  consequently  no  friction  between 
them,  and  the  system  rolls  down  a  rough  inclined  plane. 

In  this  case,  if  we  neglect  the  mass  of  the  outer  shell,  the 
inner  sphere  acts  just  as  if  it  slid  down  the  plane,  because, 
since  there  is  no  friction  between  it  and  the  shell,  as  the  shell 
rolls  it  slips  around,  and  therefore  tli°  equation  of  motion  is 

ar 

M  being  the  mass  of   the  sphere,  which  is  s,o  large  that   the 
mass  of  the  outer  shell  is  negligible  in  comparison. 

If,  however,  the  shell  and  sphere  were  united,  the  system 
would  roll  down,  and  then  the  equation  of  motion  would  be 


M 


dt'^ 


:  J/|-^sin  a. 


And  the  times  occupied  in  rolling  a  given  distance  in  the  two 
cases  would  be  to  one  another  as  Vs  :  V7. 


lere- 


MOTION   OF   A   FREE   BODY. 


I5t 


In  the  case  of  a  cylinder  surrounded  by  a  cylindrical  shell, 
gravity  would  be  diminished  to  two-tliirds  of  its  value,  and  the 
times  occupied  in  rolling  a  given  distance  would  be  as  V2  :  V3 
under  similar  circumstances. 


ugh 
two- 


4-  To  determine  whether  a  sphere  is  hollow  or  solid  by  roll- 
ing it  down  a  rough  plane.  This  could  be  done  by  observing 
the  space  passed  over  in  a  given  time,  and  by  calculating  the 
moments  of  inertia  and  forming  the  equations  of  motion  (i) 
on  the  supposition  of  a  solid  body ;  (2)  on  the  supposition  of 
a  shell  of  radii  a,  b. 

5.  A  homogeneous  heavy  sphere  rolls  down  within  a  rough 
spherical  bowl ;  it  i^,  required  to  determine  the  motion. 


)oth 
^een 

the 
use, 
hell 


the 


em 


wo 


.A£ 


Fig.  61. 


I   .  1 


iii! 


lr.ll 


l\' 


I 


ii 


iV' 


S': 


r^ 


I 


^ 


« 


152 


RIGID   DYNAMICS. 


Let  the  radius  of  the  spherical  bowl  (Fig.  61)  be  d,  and  of  the 
sphere,  a ;  and  let  the  sphere  start  with  AP  coincident  with  BQ. 
Then,  at  time  t,  the  circumstances  are  as  shown  in  the  figure. 


Let 


G)=  angular  velocity  about  P, 


OCP  =  <f>,  OM=x, 

DPA-.=  d,  PM^y, 

BCO  =  a, 
then  will  x={b—a)s\n^, 

and  y=^b  —  {b—a)cos<l). 

And  the  equations  of  motion  are 

M—^  =  —  y?  sin  6  +  Fcos  6, 
M—^  —  R  cos  (f>-\- F sin  (f)  —  m£; 


(I) 
(2) 


F  being  the  friction,  and  R  the  reaction  at  the  point  D,  acting 
in  the  directions  indicated  by  the  arrows. 


dt  dt  dt     dt 


Moreover, 
MPA  being  the  exterior  angle  at  P,  and 

a       dt 


(3) 


Along   with   the   foregoing    relation   we   have,   also,    taking 
moments  about  P, 

(4) 


MK'^—^F-a. 


dt 

It  is  then  easy  to  find  R  and  F  by  taking  the  values  of  x  and  p, 
and  differentiating  twice  and  substituting  in  (i)  and  (2). 


of  the 
\xBQ. 
ire. 


(2) 
acting 


(3) 
taking 

(4) 
and  y, 


MOTION    OF   A   FREE   BODY. 
It  will  be  found  on  reduction,  that 


153 


w/r, 


and 


R=  ^(i;  cos  <^- 10  cos  a), 
(^-«)(^J  =  -W(cos  </.-cos  a). 


From  this  latter  expression,  by  differentiation,  we  -et 


dfi         7    d-a 


sin  0, 


and  if  <f>  becomes  small,  this  gives  the  time  of  a  small  oscillation 
of  a  sphere  .  ithin  a  spherical  bowl.     For 


</)  =  0, 


represents  a  motion  of  oscillation  of  which  the  periodic  time  is 

It  may  also  be  noticed  that  the  pressure  on  the  bowl  vanishes 
when  cos  (^  =  -1^  cos  a. 

If  BOB^  were  completed  and  the  sphere  supposed  to  rotate 
about  C  with  angular  velocity  sufficient  to  keep  the  smaller 
sphere  at  the  top,  the  pressure  against  the  outer  sphere  and  the 
conditions  of  equilibrium  can  at  once  be  found  from  the  rela- 
tions already  obtained,  which  also  furnish  a  solution  to  the 
following  instructive  problem  : 

6.  A  perfectly  rough  ball  is  placed  within  a  hollow  cylindrical 
garden  roller  at  the  lowest  point,  and  the  roller  is  then  drawn 
along  a  level  walk  with  a  uniform  velocity  V.  Show  that  the 
ball  will  roll  quite  round  the  interior  of  the  roller  if  V^  be  > 
S^  g{b-a),  a  being  the  radius  of  the  ball,  and  g  of  the  roller. 


m 


|i     i: 


Pi: 


(i 


rii;  'f 


;■■,!  I-   I' 


154 


RIGID   DYNAMICS. 


7  A  uniform  straight  rod  slips  clown  in  a  vertical  plane 
between  two  smooth  planes,  one  horizontal,  the  other  vertical ; 

find  the  motion. 

Let  OX,  OY  he  the  horizontal  and  vertical  planes,  and  let 
the  rod  starting  from  its  upper  position  when  /=o  assume  the 
position  AB  at  time  /,  as  in  Fig.  62. 


Fig.  62. 


Then  we  have  two  reactions  at  the  points  A  and  B,  and  the 
weight  Mg  acting  at  the  centre  of  gravity,  C. 

s'o  that  if  x,y  be  the  coordinates  of  C,  and  6  the  angle  of 
inclination  of  the  rod  AB  to  the  horizontal,  v/e  get 


rfi 


MOTION   OF   A   FREE   BODY. 


155 


ane 
cal ; 

let 
the 


\^ 


o 


d  the 
gle  of 


and  x=acos,d, 

y  =  a  sin  6,  J 

where  2  rt-  is  the  length  of  the  rod. 

Also,  taking  moments  about  the  centre  of  gravity,  we  would 
have 

and  we  may  suppose  that  6  is  initially  equal  to  «. 

These  four  relations  give  a  complete  solution  of  the  problem. 

It  will  be  found  that  the  rod  leaves  the  vertical  plane  when 
sin  ^  =  I  sin  cc,  and  then  the  motion  becomes  changed,  the  rod 
moving  with  a  constant  horizontal  velocity  along  the  horizontal 

plane  equal  to  \\— — -,  until  it  finally  drops  and  lies  in  the 

3 
plane. 

The  problem  may  also  be  solved  by  aid  of  the  principle  of 

energy. 

8.  A  circular  disc  capable  of  motion  about  a  vertical  axis 
through  its  centre  perpendicular  to  its  plane  is  set  in  motion 
with  angular  velocity  12.  A  rough  uniform  sphere  is  gently 
placed  on  any  point  of  the  disc,  not  the  centre ;  prove  that  the 
sphere  will  describe  a  circle  on  the  disc,  and  ♦•hat  the  disc  will 

revolve  with  angular  velocity  ^— — ^- :  •  H,  where  MB  is 

^   7M/c^  +  2inr^ 

the  moment  of  inertia  of  the  disc  about  its  centre,  m  is  the 
m.ass  of  the  sphere,  and  r  is  the  radius  of  the  circle  traced  out. 

9.  A  homogeneous  sphere  is  placed  at  rest  on  a  rough 
inclined  plane,  the  coefficient  of  friction  being  jjl  ;  determine 
whether  the  sphere  will  slide  or  roll. 

10.  A  homogeneous  sphere  is  placed  on  a  rough  table,  the 
coefficient  of  friction  being  fi,  and  a  particle  one-tenth  of  the 
mass  of  the  sphere  is  attached  to  the  extremity  of  a  horizontal 


156 


RIGID   DYNAMICS. 


Ill 


in' 


■i  f 


J  I 


Mr 


m 


J 


■  V  1       M 

■  ■  1 :    ' , 

diameter.     Show  that  the  sphere  will   begin  to  roll   or  slide 

II  iin...   ...:ii   1- ic  II 


according  as  /4>or< 


What  will  happen  if  u  — 


10V37 
81.    Impulsive  Actions.     Motion  of  a  Billiard  Ball. 


10V37 


The  complex  motions  of  a  homogeneous  sphere  moving  on  a 
rough  horizontal  plane  are  well  illustrated  in  the  game  of 
billiards,  where  an  ivory  sphere  is  struck  by  a  cue  and  made 
to  perform  evolutions  that  seem  to  the  unscientific  little  short 
of  marvellous. 

In  the  general  case  the  course  which  the  billiard  ball  takes 
depends  on  the  initial  circumstances,  that  is  to  say,  on  the  way 
in  which  it  is  struck  by  the  cue  ;  and  the  motion  is  made  up  of 
both  sliding  and  rolling,  so  that  the  centre  of  the  ball  moves 
in  a  portion  of  a  parabola  until  the  sliding  motion  ceases,  when 
it  rolls  on  in  a  straight  line.  If  struck  so  that  the  cue  is  in  the 
same  vertical  plane  with  the  centre  of  the  sphere,  then  the 
motion  is  purely  rectilinear;  which  is  also  the  case  if  the  cue  is 
held  in  a  horizontal  position. 

It  may  also  happen  that  if  the  ball  be  struck  by  the  cue  at  a 
certain  oblique  inclination  to  the  table,  its  path,  after  sliding 
ceases,  will  be  opposite  to  the  horizontal  direction  of  the  stroke, 
and  it  will  roll  backwards. 

For  a  complete  solution  of  the  problem,  then,  we  should 
know  the  direction,  intensity,  and  point  of  application  of  the 
blow  struck  by  the  cue,  so  that  the  velocity  of  translation  of 
the  centre  of  gravity  is  known,  and  the  initial  angular  velocity. 

82.  In  ordinary  blows,  the  initial  value  of  the  rolling  friction 
will  be  very  small  compared  with  the  sliding  friction,  so  that  at 
the  beginning  the  former  may  be  neglected,  and  the  equations 
of  motion  for  sliding  found  in  the  following  way. 

Let  the  plane  in  which  the  centre  of  the  ball  moves  be  the 
plane  oi  xy,  so  that  {x,  y,  —a)  are  the  coordinates  of  the  point 
of  contact  at  time  t.  Let  F  be  the  value  of  the  sliding  friction, 
and  /Q  the  angle  it  makes  with  the  axis  of  x. 


MOTION   OF  A   FREE   BODY. 


157 


Then  evidently  the   pressure   on    the   table   is  equal  to  the 
weight  of  the  ball,  so  that  R  =  Mg  and  F=ixR. 

The  equations  of  motion  of  the  centre  of  gravity  are 


7l/^'=    -Fcos/3, 
M'^,--^    -Fsin(3, 

dt" 

M^^=o  =  R-Mg. 


For  rotation  about  the  centre  of  gravity  we  have 


^^=_rt/rsin/3, 


dt 
dt 
dt 


aF  cos  /3, 


'  Si 


^3  =  ^3' 

where  u^,  Vq  are  the  axial  components  of  the  initial  velocities 
of  the  centre  of  gravity,  and  fl^,  fig'  ^3  ^^^  ^^^  initial  angular 
velocities  about  axes  through  the  centre  of  gravity. 

The  above  give  a  complete  solution  of  the  motion  during 
sliding  which,  however,  in  the  case  of  an  ordinary  billiard  ball, 
lasts  but  for  a  small  fraction  of  a  second. 

83.  At  the  instant  the  ball  is  struck  by  the  cue  the  hnpidsive 
equations  will  evidently  be  formed  as  follows. 


158 


RIGID    DYNAMICS. 


I'      'Jf 


,  I 

if 
( 


f 


l.^:•^T! 


r 


Let  Q  be  the  value  of  the  blow  struck  by  the  cue,  and  «  the 
angle  the  cue  makes  with  the  table  ;  also,  let  F  be  the  impulsive 
value  of  friction  at  the  instant  of  striking,  and  ^  the  angle 
which  it  makes  with  the  axis  of  x. 

Then,  the  axes  being  chosen  as  in  the  preceding  problem, 
and  the  line  and  angular  velocities  being  denoted  as  formerly 
by  Uq,  Vq,  Hj    ^^,  1>     we  have 

'   ,4/,^    z  Q  cos  a  —  Fcos  /3, 
\Mvq=  -Fsin/3, 

AD.-^=  —Qh  sin«— rt/^sinyS, 
Ail^=     Qk         -\-aF  cos  ^, 
,  AD,^=  —  Q/i  cos  a, 

where  /i  is  the  horizontal  distance  from  the  centre  of  the  ball  to 
the  vertical  plane  containing  the  line  of  blow,  and  k  is  the  per- 
pendicular on  the  line  of  blow  from  the  point  where  /i  meets  the 
vertical  plane  containing  that  line.  And  the  impulse  on  the 
table  must  be  equal  to  Q  sin  a.  See,  Theorie  mathhnatique 
des  effcts  du  jeu  de  billard,  par  G.  Coriolis,  Paris,  1835. 

84.    Impulsive  Actions.     Free  Body.     Illustrative  Examples. 

1.  A  uniform  rod  is  lying  on  a  smooth  horizontal  table  and 
is  struck  at  one  end  in  a  direction  perpendicular  to  its  length. 
Determine  the  motion. 

What  if  it  be  struck  at  the  centre,  or  at  the  centre  of  per- 
cussion for  a  rotation-axis  through  one  end  of  the  rod } 

2.  Two  uniform  rods  of  equal  length  are  freely  hinged 
together  and  placed  m  a  straight  line  on  a  smooth  horizontal 
plane.  The  system  is  then  struck  at  one  end  in  a  direction 
perpendicular  to  its  length.  Examine  the  motion  initially  and 
subsequently,. 


MOTION   OF   A   FREE   BODY. 


159 


l> 


Here,  the  circumstances  are  a  little  more  complicated  than 
in  the  preceding  problem,  so  that  it  is  well  to  form  the  equa- 
tions of  motion  of  the  two  rods  separately. 

Let  in  be  the  mass  of  each  rod,  2  a  the  length,  C,  C  the 
centres  of  gravity,  v,  v'  the  velocities  of  translation  of  C,  C\ 
and  o),  w'  the  angular  velocities. 

Then  if  (9,  O'  be  the  instantaneous  centres  so  that  CO=a. 
and  C'O'  =x',  we  get 

cox  =  7', 
(o'x'  =v', 


and 


(a  —x)co  =  {a  -{-x')q)'. 


And  if  Q  be  the  blow,  and  R  the  react'.  ;  a    the  free  hinge, 
the  equations  of  motion  of  the  two  rods  ar . 


mv=Q-\-R, 
ma(o     ^     o 


mv'  =  R, 


mao^ 


=  K 


per- 


from  which  it  will  be  found  that 

6)  =  2  ft)', 

and  the  initial  velocity  of   the  end   struck   is   four  times  that 
of  the  other  end. 


3.  Three  uniform  and  equal  rods  AB,  BC,  CD  arc  arranged 
as  three  sides  of  a  square  having  free  hinges  at  B  and  C;  the 
end  A  is  struck  in  the  plane  of  the  rods  and  at  right  angles 
to  AB  by  a  blow  Q.  Determine  the  motion,  and  show  that 
the  initial  velocity  of  A  is  nineteen  times  that  of  D. 


•  ,-S.v 


i6o 


RHAD   DYNAMICS. 


41 


ii' 


This  is  solved   in  the  same  way  as  the  former  problem  by 

considering  each  portion  separately.     Thus,  if  /v  be  the  reaction 

of  B,  we  ha'^e 

mv  =Q  +  R, 


maw 


and 


3 
tax—v. 


Q-R, 


Also,  if  R'  be  the  reaction  at  Q 

R  —  R'  =  m{a  —x)w  =  m{a  4-.t-')<u', 

since  the  displacements  of  B  and  C  are  equal  and  in  the  same 
direction. 

And  for  CD,  vn''=R', 


ma(o 


=  R'. 


4.  If  in  the  preceding  problem  BC  be  a  thin  string  whose 
mass  is  negligible,  show  that  the  initial  velocity  of  A  will  be 
seven  times  that  of  D. 

This  is  evident,  for  R  —  R'. 

5.  Two  equal  uniform  rods  AB,  BC,  freely  jointed  at  B,  are 
placed  on  a  smooth  horizontal  table  at  right  angles  to  one 
another,  and  a  blow  is  applied  at  A  perpendicular  to  AB ; 
prove  that  the  initial  velocities  of  A,  C  are  as  8  to   i. 

6.  Four  equal  uniform  rods  AB,  BC,  CD,  DE,  freely  jointed 
at  B,  C,  D,  are  laid  on  a  horizontal  table  in  the  form  of  a  square, 
and  a  blow  is  applied  at  A  at  right  angles  to  AB  from  the  inside 
of  the  square ;  prove  that  the  initial  velocity  of  A  is  79  times 
that  of  E. 

7.  Three  equal  inelastic  rods  of  length  a,  freely  hinged 
together,  are  placed  in  a  straight  line  on  a  smooth  horizontal 
plane,  and  the  two  outer  ends  are  set  in  motion  about  the  ends 


MUTIUN    OK   A    FREI"    BODY. 


I6l 


of  the  middle  rod  with  tMiiud  but  opposite  anguhir  velocities  (tw)  ; 
show  that  after  impact  the  triangle  formed  by  the  three  will 
move  cri  with  a  velocity  I  am. 

8.  Four  equal  rods  freely  jointed  toj^ether  so  as  to  form  a 
square  are  moving  with  given  velocity  in  the  direction  of  a 
diagonal  of  the  square,  on  a  smooth  horizontal  plane.  If  one 
end  of  this  diagonal  impinge  directly  on  an  inelastic  obstacle, 
find  the  time  in  which  the  rods  will  be  in  one  straight  line, 

9.  Four  equal  uniform  rods  each  connected  by  a  hinge  at 
one  extremity  with  the  middle  point  of  the  rod  next  in  order, 
initially  form  a  square  with  produced  sides,  and  are  in  motion 
with  a  given  velocity  in  direction  parallel  to  one  of  the  rods. 
If  an  impulse  be  given  at  the  free  extremity  of  this  rod,  and 
the  centre  of  inertia  of  the  system  be  thereby  reduced  to  rest, 
find  the  initial  angular  velocities  of  the  four  rods,  and  prove 
that  these  angular  velocities  remain  unchanged  during  the  sub- 
sequent motion. 

10.  A  lamina  in  the  form  of  an  ellipse  is  rotating  in  its  own 
plane  with  angular  velocity  w  about  a  focus.  Suddenly  this 
focus  is  freed  and  the  other  fixed.  Find  the  velocity  about  the 
second  focus. 

M 


\  I 


i. '' 


r ' ! 


■(  <  i    I 


CHArricR  X. 


THE   uYROSCOI'K 


85.  This  instrument,  to  which  reference  has  already  been 
made  in  connection  with  motion  about  a  fixed  point,  consists 
essentially  of  a  wheel  which  is  put  in  rotation  within  an  outer 
ring :  the  latter  being  provided  with  knife  edges  and  other 
arrangements  whereby  the  whole  mass  may  be  experimented 
upon  while  the  wheel  is  kept  in  motion. 

A  type  of  gyroscope,  known  as  Foucaulfsy  is  shown  in  Fig. 
63,  and  also  more  in  detail  in  Figs.  65  and  &^. 


: 


iJ         l: 


Fig.  63. 


It  is  made  of  a  disc,  turned  to  offer  the  least  resistance  to 
the  air,  which  can  be  made  to  rotate  with  great  speed  (from 
two  hundred  and  fifty  to  five  hundred  times  per  second)  about 
an  axis  through  its  centre  of  gravity. 

This  is  done  by  means  of  the  wheclwork  motor  (driven  by 
hand)  shown  in  Fig.  64,  which  is  geared  up  at  the  top  to  the 
small  toothed  cog-wheel  seen  in  Fig.  63,  at  the  left-hand  side 
of  the  disc,  on  the  axis  of  the  gyroscope,  and  within  the  outer 


rmg. 


162 


Fig. 


Tin;    CYKOSCOPK. 


165 


The  axis  oi  rotation  is  of  course  movable  in  the  outer  rinj;, 
and  this  latter  is  provided  with  two  knife  ed^cs  which  should 
be  exactly  in  the  prolon<;ation  of  a  line  passing  through  the 
centre  of  gravity  and  perpendicular  to  the  rotation  axia. 


Fig.  64. 

Four  movable  masses,  two  within  the  ring,  and  two  outside, 
Fig.  65,  are  used  to  adjust  the  instrument  in  two  perpendicular 
planes,  so  that  the  centre  of  gravity  of  the  system  will  be  in  the 
line  of  the  knife  edges. 

It  is  quite  a  diflficult  matter  to  perform  this  adjustment,  which 
must  be  exact ;  since  the  slightest  deviation  of  the  position  of 
the  centre  of  gravity  from  this  line  destroys  the  value  of  the 
results  obtained  in  the  pendulum  experiment. 

The  readiest  way  to  adjust  the  gyroscope  is  to  let  it  oscillate, 
un-Ier  the  action  of  gravity,  about  the  knife  edges,  the  centre 
of  gravity  being  arranged  at  first  to  fall  below  the  line  of  the 
knife  edges  (by  properly  altering  the  positions  of  the  movable 
masses) ;  and  then,  by  slight  variations  of  these  positions,  to 
bring  the  centre  of  gravity  up  until  the  oscillations  about  the 
knife-edge  axis  are  made  in  from  eight  to  ten  seconds :  the  line 
of  the  knife  edges  is  in  that  case  infinitely  close  to  the  centre  of 
gravity  and  the  equilibrium  nearly  neutral. 


164 


RIGID   DYNAMICS. 


'f 


ir 


11    I 


86.  The  Gyroscope  moving  in  a  Horizontal  Plane  about  a 
Fixed  Point. 

The  gyroscope  being  adjusted,  the  experiment  indicated  by 
the  theory  of  Art.  72  may  easily  be  performed. 

It  is  only  necessary  to  place  the  instrument  on  top  of  the 
motor  so  that  the  wheels  are  properly  geared,  and  to  set  the 
disc  in  rapid  rotation,  taking  care  that  the  bearings  are  care- 
fully cleaned  and  oiled. 

Then,  placing  it  as  shown  in  Fig.  65,  so  that  a  small  pointed 
hook  which  is  directly  in  the  prolongation  of  the  axis  of  rota- 


Fig.  65. 


Fig.  66. 


tion  rests  on  a  little  agate  cup  at  the  top  of  an  upright  stand, 
the  instrument  is  given  a  slight  angular  displacement  bodily 
about  a  vertical  axis  passing  through  the  point  at  which  the 
hook  rests,  and  it  slowly  moves  about  the  vertical  with  an  angu- 
lar velocity  equal  to  that  found  by  the  theory  of  Art.  72. 

Moreover,  the  direction  of  motion  is  as  shown  in  Fig.  66 ; 
that  is,  the  gyroscope  moves  bodily  about  a  vertical  axis  (when 
viewed  from  above)  in  the  same  direction  as  the  disc  rotates 
when  viewed  by  an  observer  looking  towards  the  fixed  point 
about  which  the  motion  takes  place. 

Thus  there  is  a  perfect  accord  between  theory  and  exper- 
iment, and  the  truth  of  the  fundamental  equations  of  motion 
is  established. 


,H 


t: 


i 


THE   GYROSCOPE. 


I 


1 
1 


165 


Fig.  67. 


I 


t. ' 


1 66 


RIGID    DViNAMlCS. 


Ij  h,  ■■  f 


Ic-.'R 


It  may  be  observed  also  that  if  the  gyroscope  be  given  no 
initial  impulse,  but  be  merely  let  drop,  it  will  act  in  the  same 
manner  as  a  top,  and  oscillate  up  and  down  while  it  keeps  in 
motion  about  the  vertical. 

87.    To  p7'ovc  the  Rotation  of  the  Earth  Jipon  its  Axis. 

This  experiment  depends  on  the  permanency  of  the  rotation 
axis  in  space. 

A  stand  with  pendulum  is  arranged  as  shown  in  Fig.  6^. 

There  is  a  ring  suspended  by  means  of  a  fibre  without  torsion 
from  a  hook  above,  and  the  whole  being  carefully  levelled  so 
that  the  line  of  suspension  is  vertical,  the  gyroscope  is  put 
in  rapid  rotation  and  placed  in  the  ring  with  the  knife  edges 
resting  within  beds  provided  for  them  :  the  ring,  being  then 
released  by  the  small  screw  seen  at  the  right,  is  quite  free 
in  space,  and  owing  to  the  rapid  rotation  of  the  disc  the  axis  ot 
rotation  is  a  permanent  axis  and  remains  fixed  in  space. 

Hence,  while  the  earth  moves  along,  carrying  with  it  the  stand 
and  observer,  the  gyroscope  preserves  its  position  in  space  for 
some  time ;  and  if  a  long  index  be  attached  to  it  in  prolonga- 
tion of  the  rotation  axis  or  parallel  to  it,  this  index  will  have  an 
apparent  motion  from  east  to  west,  as  the  observer  is  carried 
along  with  the  earth  from  west  to  east. 

If  the  pendulum  with  the  gyroscope  were  placed  at  the  north 
pole,  it  is  evident  that  the  apparent  motion  of  the  index  would 
be  360°  in  twenty-four  hours. 

At  the  equator  there  would  be  no  apparent  motion ;  as  although 
a  permanent  axis  would  still  exist,  the  earth  would  simply  carry 
the  whole  instrument  bodily  about  the  rotation  axis  of  the  earth. 

Action  in  Any  Latitude  \. 

To  find  the  angular  velocity  of  the  gyroscope  in  any  latitude, 
let  PCF,  Fig.  68,  be  the  axis  of  rotation  of  the  earth. 

And  let  the  gyroscope  be  suspended  at  A,  in  the  tangent 
plane,  and  preferably  let  the  plane  of  rotation  of  the  disc  be  in 
the  geographical  meridian  plane. 


i 


THE   GYROSCOPE. 


i6: 


Then  the  angular  velocity  of  the  earth  about  PCF  is 
o)  =  360°  in  twenty-four  hours. 

And  this,  if  resolved  along  CA,  will  produce  a  rotation  about 
CA  equal  to  w  sin  X,  and  this  is  the  component  which  affects  the 
gyroscope  at  A. 

Since  w  is  against  the  hands  of  a  watch,  looking  towards  C 
from  P,  therefore  w  sin  \,  looking  from  A'  towards  A  or  C, 
will  be  against  the  hands  of  a  watch,  and  therefore  if  Fig.  69 


W 


represents  the  tangent  plane  at  A,  to  an  observer  at  A'  above 
the  gyroscope,  the  earth  will  move  from  west  to  east  as  indi- 
cated by  the  arrow,  and  the  apparent  motion  of  the  index 
attached  to  the  gyroscope  will  be  as  before  from  cast  to  west. 

88.  It  is  evident  also  that  the  angular  velocity  being  w  sin  \, 
if  this  be  observed  by  noting  the  time  and  the  angle  passed  over 
in  that  time,  since  w  is  known  to  be  360°  in  twenty-four  hours, 
we  get  a  method  for  finding  \,  the  latitude  of  the  place  of 
experiment. 


fff 


l68 


RIGID   DYNAMICS. 


89.    Electrical  Gyroscope. 

The  defect  of  Foucault's  gyroscope  being  that  it  does  not 
keep  up  its  motion  long  enough  to  give  marked  results  in  the 
pendulum  experiment,  an  electrical  gyroscope  has  been  devised 
by  Mr.  Hopkins,  who  gives  a  description  of  bis  instrument  in  the 
Scientific  American  of  July  6,  1878,  and  also  in  his  recent  text- 
book on  Physics.     His  instrument  is  shown  in  Fig.  70. 


'1' 


1    t. 
I  I 


Fig.  70 


The  rectangular  frame  which  contains  the  wheel  is  supported 
by  a  fine  and  very  hard  steel  point,  which  rests  upon  an  agate 
s'up  in  the  bottom  of  a  small  iron  cup  at  the  end  of  the  urr 
thai  is  supported  by  the  standard.  The  wheel  spindle  turns  on 
carefully  made  'teel  points,  and  upon  it  are  placed  two  cams, 
.)iio  :i-  each  end,  which  operate  the  current-breaking  springs. 


THE   GYROSCOPE. 


169 


;S  not 
in  the 
;vised 
in  the 
t  text- 


pported 
n  agate 
:he  urn 
urns  on 
0  cams, 


uigs. 


The  horizontal  sides  of  the  frame  are  of  brass,  and  the  ver- 
tical sides  are  iron.  To  the  vertical  sides  are  attached  the 
cores  of  the  electro-magnets.  There  are  two  helices  and  two 
cores  on  each  side  of  the  wheel,  and  the  wheel  has  attached  to 
it  two  armatures,  one  on  each  side,  which  are  arranged  at 
right  angles  to  each  other.  The  two  magnets  are  opp^  >itely 
arranged  in  respect  of  polarity,  to  render  the  instrument  astatic. 

An  insulated  stud  projects  from  the  middle  of  the  lower  end 
of  the  frame  to  receive  an  index  that  extends  nearly  to  the 
periphery  of  the  circular  base  piece  and  moves  over  a  graduated 
semicircular  scale.  An  iron  point  project:,  from  the  insulated 
stud  into  a  mercury  cup  in  the  centre  of  the  base  piece,  and  is 
in  electrical  communication  with  the  platinum  pointed  screws 
of  the  current  breakers.  The  current-breaking  springs  are  con- 
nected with  the  terminals  of  the  magnet  wires,  and  the  magnets 
are  in  electrical  communication  with  the  wheel-supporting  fraine. 
One  of  the  binding  posts  is  connected  by  a  wire  with  the  irer- 
cury  in  the  cup,  and  the  other  is  connected  with  the  stand- 
ard. A  drop  of  mercury  is  placed  in  the  cup  that  contains  the 
agate  step  to  form  an  electrical  connection  between  the  iron 
cup  and  the  pointed  screw. 

The  current  breaker  is  contrived  to  make  and  break  the 
current  at  the  proper  instant,  so  that  the  full  e'  Jct  of  the  mag- 
nets is  realized,  and  when  the  binding  posts  n  connected  with 
four  or  six  Bunsen  cells  the  wheel  rotates  at      iiigh  velocity. 

The  wheel  will  maintain  its  plane  of  rotat  n,  and  when  it  is 
brought  into  the  plane  of  the  meridian,  the  index  will  appear  to 
move  slowly  over  the  scale  in  a  direction  t  ntrary  to  the  earth's 
rotation,  but  in  reality  the  earth  and  the  scale  with  it  move 
from  west  to  east,  while  the  index  remains  nearly  stationary. 


90.    FesscTs  Gyroscope. 

Another  most  useful  and  instructive  for 


of  gyroscope   is 


that  known  as  Fessel's,  which  is  represented  ni  Fig.  71. 

"  (2  is  a  heavy  fixed  stand,  the  vertical  shaft  of  which  is  a 


i 


'S  f  ■■ 

i  1 

f  'i 

1 
!         I 

:  ■    1 

1 

f 

ilii 


iii 


'i' 


170 


RIGID   DYNAMICS. 


cylinder  bored  smoothly,  in  which  works  a  vertical  rod  CC,  as 
far  as  possible  without  friction,  carrying  at  its  upper  end  a  small 
frame  BB'.  In  BB'  a  horizontal  axis  works,  at  right  angles  to 
which  is  a  small  cylinder  /?,  with  a  tightening  screw  7/,  through 
which  passes  a  long  rod  G(7',  to  one  end  of  which  is  affixed  a 
large  ring  AA',  and  along  wliich  slides  a  small  cylinder  carrying 
a  weight  IV,  which  is  capable  of  being  fixed  at  any  point  of  the 


rod  ;  and  so  that  it  may  act  as  a  counterpoise  to  the  ring,  or  to 
the  ring  and  any  weight  attached  to  it.  An  axis  AA'  works  on 
pivots  in  the  ring,  in  the  same  straight  line  with  GG'  ;  to  AA' 
a  disc,  or  sphere,  or  cone,  or  any  other  body,  can  be  attached, 
and  thus  can  rotate  about  AA'  as  its  axis  ;  to  the  body  thus 
attached  to  A  A'  a  rapid  rotation  can  be  given,  either  by  means 
of  a  string  wound  round  A  A'  or  by  a  machine  contrived  for  the 
purpose  when  A  A'  and  its  attached  body  are  applied  to  it.  It 
is  evident  that  the  counterpoise  JV  can  be  so  adjusted  that  the 
centre  of  gravity  of  the  rod,  the  ring,  the  attached  body,  and 
the  counterpoise,  should  be  in  the  axis  BB' ;  or  at  any  point  on 
either  side  of  it  ;  that  is,  /i  may  be  positive,  or  be  equal  to  o,  or 
may  be  negative.  Also  by  fixing  BB'  in  the  arm  of  CC  which 
carries  it,  the  inclination  of  the  rod  GG'  to  the  vertical  may  be 
made  constant ;  that  is,  0  may  be  equal  to  6^  throughout  the 
motion.  When  the  counterpoise  is  so  adjusted  that  the  centre  of 
gravity  of  the  rod  GG'  and  its  appendages  is  in  CC',  then  //  =  o, 
or,  what  is  equivalent,  7/1/1^  =  o."  (Price,  Calculus  ;  vol.  iv.) 
It  is  evident  that  with  such  an  instrument,  with  its  various 


re,  as 
I  small 
gles  to 
trough 
fixed  a 
irrying 
of  the 


g,  or  to 
orks  on 
to  AA' 
ttachcd, 
dy  thus 
'  means 

for  the 
)  it.  It 
that  the 
)dy,  and 
ooint  on 
to  o,  or 
r'  which 

may  be 
lOut  the 
centre  of 
m  h  =  o, 
»1.  iv.) 
i  various 


THE   GYROSCOPE. 


171 


adjustments,  all  the  motions  about  a  fixed  point  can  be  fully  dis- 
played and  examined ;  and  the  results  already  obtained  in  the  case 
of  the  top  (Art.  (yG)  and  the  gyroscope  (Art.  72)  thereby  shown. 

91.  Another  form  of  gyroscope  worthy  of  notice  is  that  first 
constructed  by  Professor  Gustav  Magnus  of  l?crlin,  and  de- 
scribed by  him  in  Poggendorff's  Annalcn  dcr  PJiysik  uud 
Clicmic,  vol.  xci.,  pp.  295-299.  The  instrument  consists  of  two 
rings  and  discs  such  as  AA\  Fig.  71,  connected  by  a  rod  sup- 
ported in  much  the  same  way  as  the  rod  GQ  in  Fessel's  gyro- 
scope. There  is  a  binding-screw  at  B,  to  arrest,  when  so 
desired,  motion  about  the  horizontal  axis  BIV ,  and  also  a  short 
rod  projecting  horizontally  from  the  upper  part  of  the  vertical 
axis  CC  by  which  the  motion  about  that  axis  may  be  accel- 
erated, retarded,  or  completely  arrested  at  will.  By  means  of 
two  cords  wound  round  their  axes  and  simultaneously  pulled  off, 
the  discs  can  be  put  in  rapid  rotation  with  nearly  equal  veloci- 
ties either  in  the  same  or  in  opposite  directions.  The  follow- 
ing phenomena  are  exhibited  by  this  apparatus : 

If  tlie  connecting  rod  be  supported  midway  between  the 
discs,  and  if  the  discs  be  made  to  rotate  rapidly  with  equal 
velocities  in  the  same  direction,  and  no  weight  be  suspended 
at  W  (Fig.  71),  the  connecting  rod  will  remain  at  rest.  If  a 
weight  be  suspended  at  W,  the  rod  and  discs  will  slowly  rotate 
about  the  vertical  axis  CO .  If  the  motion  round  the  vertical 
axis  be  accelerated,  the  loaded  end  of  GG  will  rise,  if  the 
horizontal  rotation  be  retarded,  the  loaded  end  will  sink.  If 
the  binding-screw  be  tightened  so  as  to  arrest  this  rising  or 
sinking,  the  rotation  about  the  vertical  axis  will  also  cease,  to 
commence  again  as  soon  as  the  binding-screw  is  loosened. 

If  the  discs  rotate  with  equal  velocities  in  opposite  directions, 
the  loaded  end  of  GG'  will  sink.  If  the  connecting-rod  be  sup- 
ported at  a  point  nearer  to  one  disc  than  to  the  other,  and  the 
discs  be  made  to  rotate  with  equal  velocities  in  opposite  direc- 
tions, the  instrument  will  still  be  found  extremely  sensitive. 


'f-  — 


ii 


■  I  4 1 


'!)  1 


NOTE  ON  THE  PENDULUM  AND  THE  TOP. 


^'n> 


I.    In  Art.  35,  pp.  47  to  49,  we  have  found  the  equation 
(//^  +  /f'^)  (  - )  =2  ^//  (cos  6  —  cos  «), 
or,  as  it  may  be  written  (sec  page  50), 

H~r)  =  2^ (cos  c/— cos  a) 


(0 


for  the  oscillations  of  a  rigid  body  about  a  fixed  horizontal  axis, 
and  have  applied  it  to  the  case  of  a  pendulum  making  extremely 
sma!'  oscillations.  We  shall  here  consider  the  general  case, 
when  the  arc  of  the  oscillations  is  not  necessarily  small. 


Let 


and 

Differentiating, 


cos  ^  — cos  «=(i  — cos«)  cos^  (/>. 
.•.   I  — cos  ^  =  (1  — cos«)  sin^  ^ 

cos  6  =  cos^0  +  cos  a  sin^  ^. 


(ii) 


sm  (/— -  =2  sm  4>  cos  (6(1  —cos  ct)~. 


:.  (i+cos^) 


de\^ 

(it 


=  4(cos^  — ccjsa),  \-T.] 


SubstitAiting  in  (i), 


l\-j-\  =^(i-sin2|asin2<^). 


172 


NOTE  ON  THE  PENDULUM  AND  THE  TOP. 


173 


(i) 


(ii) 


Let 
and 


«2=sin*^  a, 


(iii) 


and  (ii)  becomes 


(iv) 


\dt) 

J"*  d^ 

an  elliptic  integral  of  the  first  kind. 

.'.  ^  =  am(i//), 

cos  ^  =  cn2  (i;/)  +  cos  a  ^v?{yi). 

Equation  (ii)  may  be  written  in  the  form 

sin  ^^  =  sin?,  f<sinrf>; 
consequently  (iv)  may  be  written  in  the  form 

sin  .]  0  =  sin  \  u  sn  vt. 

This  equation  determines  the  position  of  the  pendulum  at  any 
given  instant,  and,  by  inversion,  the  dmes  at  which  the  pendulum 
is  in  a  given  position. 

If  The  the  period  of  the  pendulum,  i.e.  the  length  of  time 
required  for  the  pendulum  to  make  a  double  swing  through  the 
arc  2  a, 


Integrating  and  writing  ^  for  1^  and  sin  |  «  for  k, 

r=27r^(^) 1 1  +(|)2(sin|)2  +  (^ll^y  (sin \ af 


{y} 


ii 


M 


'      I 


I  : 
I      I 


ll'i 


'74 


RIGID   DYNAMICS. 


2.    In  Art.  67,  p.  118,  wo  have  found  the  ccpiation 


( 


A  sin  ^y  Y=  (cos  (9„-cos  e)\2AMg/i  sin2 9 

-  C-//2(cos  ^„-cos  e)\,         (n) 

for  the  nutation  oscillations  of  a  top  spinning  about  a  fixed  point, 
and  in  Art.  68  we  have  determined  the  approximate  period  of 
small  oscillations.  The  period  of  oscillations  of  any  magnitude 
anu  the  value  of  d  at  any  given  instant  may  be  determined  as 
follows : 

Let  A  =  Jr(//'  +  P)  =  jr/i/,         (see  page  50) 

and  2  A Mgh  si n^  ^  -  C  V  (^os  ^0  -  cos  (9) 

=  2  A  Mgh  (cos  e  -  cos  ^i)(cosh  7  -  cos  (9), 
which  requires  that 


cos  ^i  +  cosh  7  = 


2  AMdi 


and 


cos  ^,.  cosh  7=^'^^'"^^^" -I. 

^  '       2  AMo-h 


Substituting  in  {a\  that  equation  becomes 

sin^^,~j  =2^i^(cos^o-cos^)(cos^-cos^i)(cosh7-cos^).  (i) 

Let       cos  Bq  —  cos  6= (cos  ^^  —  cos  ^j^ )  cos^t. 
.-.  cos^  — cos^i  =  (cos^o  — cos^i)sin2T, 
and  cos^  =  cos^iCosV  +  cos^ySin2T.  (2) 

Differentiating, 

—  sin  ^  — -  =  2  sin  T  cos  t  (cos  6^  —  cos  6,)—- 

.:  ('sin6'^Y=4(cos^,-cos^)(cos^-cos6'jC^ 
Substituting  in  ( i ), 
Hy.)  = ']  c?"  fcosh 7-cos  ^1— (cos ^^--cos 6^)  sin^rf 


a 


<> 


(n) 


(2) 


^• 


Let 
and 


NOTE  ON  THL  PENDULUM  AND  THE  TOP.     ,75 

1/1  zi  \  I         cos  0,)  — cos  6^1    .   .,    1 

=  \  ^'•(cosh  7-C0S  ^1)1-       ,  J  siii-'t 

^    i        cosh  7 -cos 6^1  I 

=^(co.h» .  7-co.s»l  0,)} ,  -  -^^^^^^^^  X  ,„,v{. . 


AC"  = 


cosh^^7-C()s^^<9i' 
/i/2=^'-(cosh2.]7-cos2.](9j). 

lit)  ' 


»/ii   V  ( I  —  K^  sinV 


V  ( I  -  «"'*  sinV) 

an  elliptic  integral  of  the  first  kind. 

.'.  T=am(i^/), 
and  (2)  becomes 

cos  Q  =  cos  0^^  en-  (i^/)  +  cos  d^  sn^  (^ /), 


(4) 


thus  determining  the  inclination  of  the  axis  of  the  top  to  the 
vertical  at  any  given  instant. 

The  period  of  a  complete  oscillation  will  be 


y._4  ri itr 

vjo   •%/(  I  —  «''^  sin' 


V( 


'r) 


\\^(cosh-A  7  — eos-^  ^i)y  (         -  \2'4J 


I-3-5 


^'i^J'^^^-i 


(5) 


Comparing  equations  (4)  and  (5)  with  (iv)  and  (v),  it  wi:l  be 
seen  that  the  top's  oscillations  in  nutation  are  of  exactly  the 
same  character  as  the  oscillations  of  an  ordinary  pendulum. 
Note,  however,  that  in  the  discussions  of  the  oseillations  of  the 
pendulum,  6  is  measured  from  an  initial  axis  directed  straight 
downwards,  while  in  the  discussion  of  the  motion  of  the  top,  0 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


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Sciences 
Corporation 


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176 


RIGID   DYNAMICS. 


is  measured  from  the  vertical  axis  as  initial,  so  that  0,  «,  an  J  </> 
in  the  former  discussion  should  be  replaced  by  tt  —  O,  tt  —  u,  and 
TT  — T,  to  bring  it  into  strict  conformity  of  notation  with  the 
discussion  of  the  movements  of  the  top.  On  makin<;  these 
chanfijes,  it  will  be  found  that  the  pendulum  oscillating  about  a 
fixed  horizontal  axis  is  merely  the  special  case  of  the  top  in 
which  ^o  =  7r  — «,  d^  =  Tr,  n  =  o,  and,  therefore,  7  =  0  and  ^  is 
constant. 

Equation  (4)  enables  us  to  find  the  value  of  0  at  any  given 
in.stant  /,  but  to  completely  determine  the  position  of  the  top,  it 
is  necessary  to  be  able  also  to  find  yjr.  To  do  this  requires  the 
integration  of  the  equation  (d)  of  Art.  65,  p.  1 16,  which  may  be 
reduced  to  the  integration  of  two  elliptic  integrals  of  the  third 
kind,  as  follows : 


A  Gin2^  "^f-  =  Cu  (cos  e^^  -  cos  0). 


(^) 


dt''  A  \i-\- COS  d     I— cos  (9 


cos2 1  <9,, 


A  \i  +COS  6^  cos'-^T  +  cos  ^f)  sinV 

sina.V^', 


I  —cos 6^  cos'-T  — cos  6q  sin'-T 

C0S2  1^,^ 


Cu 

A  \\  -|-cos<?j  +  (cos^Q  — cos^i)sn''^(i'/) 

sin2 1  e. 


I— cos  ^j— (cos  ^Q  —  COS  0^)sn-{vt), 


;)• 


Thus  \|r  is  expressed  as  the  difference  of  two  elliptic  integrals 
of  the  third  kind. 


MISCELLANEOUS    EXAMPLES. 


-^>o:*:c 


(^) 


1.  Find  the  principal  axes  of  a  quadrant  of  an  ellipse  at  the 
centre. 

2.  If  a  rigid  body  be  referred  to  three  rectanj^ular  axes  such 
that  A=B  and  ^{mxy)  =  o,  show  that  the  mean  principal 
moment  of  inertia  =  A. 

3.  Determine  the  position  of  a  point  O  in  a  trianjjjular  lamina, 
such  that  the  moments  of  inertia  of  AOll  BOC,  CO  A,  about  an 
axis  through  O,  perpendicular  to  the  plane  of  the  lamina,  may 
all  be  equal. 

4.  Find  the  moment  of  inertia  of  "^he  solid  formed  by  the 
revolution  of  the  curve  r  =  a{\  +cos^)  about  the  initial  line, 
about  a  line  through  the  pole  perpendicular  to  the  initial  line. 

5.  A  uniform  wire  is  bent  into  the  form  of  a  catenary.  Find 
its  moments  of  inertia  about  its  axis,  and  its  directrix. 

6.  Find  the  moment  of  inertia  of  a  paraboloid  of  revolution 
about  a  tangent  line  at  the  vertex;  the  density  in  any  plane 
perpendicular  to  the  axis  varying  as  the  inverse  fifth  power  of 
its  distance  from  the  vertex. 

7.  Find  the  moment  of  inertia  of  a  semi-ellipse  cut  off  by  the 
axis  minor  about  the  line  joining  the  focus  with  the  extremity  of 
the  axis  minor. 

8.  If  the  moments  of  inertia  of  a  rigid  body  about  three  axes, 
passing  through  a  point  and  mutually  at  right  angles,  be  equal 
to  one  another,  show  that  these  axes  are  on  the  surface  of  an 

N  I'J 


1/8 


RIGID   DYNAMICS. 


.   I 


elliptic  cone  whose  axis  is  that  of  least  or  greatest  moment 
according  as  the  mean  moment  of  inertia  is  greater  or  less 
than  the  arithmetic  mean  between  the  other  two. 

9.  Show  that  the  moment  of  inertia  of  l  regular  octahedron 
about  one  of  its  edges  is  |  cfiM,  where  a  is  the  length  of  an  edge 
and  M  is  the  mass  of  the  octahedron. 

10.  Prove  that  the  moment  of  inertia  of  a  solid  regular  tetra- 
hedron  about  any  axis  through  its  centre  of  inertia  is  M — ,  a 
being  the  length  of  an  edge. 

11.  If  /3,  7  be  the  perpendiculars  from  B  and  C  on  a  principal 
axis  at  the  angular  point  A  of  the  triangle  ABC,  show  that 

(,,2  _  ^2  _  ^2)(^2  _  ^2)  =  ^2^2  _,.  ^2^2  ^_  2  (/,2  _  ^2)^^, 

12.  Show  that  if  a  plane  figure  have  the  moments  of  inertia 
round  two  lines  in  it,  not  perpendicular  to  one  another,  equal,  a 
principal  axis  with  respect  to  the  point  of  intersection  bisects 
the  angle  between  them. 

13.  Determine  the  points  of  an  oblate  spheroid  with  respect 
to  which  the  three  principal  moments  are  equal  to  one  another. 

14.  Show  that  the  conditions  which  must  be  satisfied  by  a 
given  straight  line  in  order  that  it  may,  at  some  point  of  its 
length,  be  a  principal  axis  of  a  given  rigid  body,  is  always  satis- 
fied if  the  rigid  body  be  a  lamina  and  the  straight  line  be  in  its 
plane,  unless  the  straight  line  pass  through  the  centre  of  inertia. 

15.  If  a  straight  line  be  a  principal  axis  of  a  rigid  body  at 
every  point  in  its  length,  it  must  pass  through  the  centre  of 
inertia  of  body. 

16.  Assuming  that  the  radius  of  gyration  of  a  regular  poly- 
gon of  ;/  sides  about  any  axis  through  its  centre  of  inertia  and 
in  its  own  plane,  is 

^-AMl  2-hCOS 

11 


ifM; 


-fcos— yf  I- 


^°^?)}' 


where  c  is  the  length  of  any  side ;  find  the  radius  of  gyration 


MISCELLANEOUS    EXAMI'LES. 


179 


less 


.. '^ 


of  a  circular  disc  about  a  line  through  any  point  in  its  circum- 
ference and  perpendicular  to  its  plane,  and  show  thai  it  is  e(|ual 
to  the  radius  of  a  circular  ring  about  a  tangent. 

17.  The  locus  of  a  point  such  that  the  sum  of  the  moments 
of  inertia  about  the  principal  axes  through  the  point  is  constant, 
is  a  sphere  whose  centre  is  the  centre  of  inertia  of  the  body. 

18.  li  A,  B,  C  be  the  moments  of  inertia  of  a  'oody  about 

principal  axes, 

A  cos^  a^  B  cos^  ^  •\- C  cos^  7 

will  be  the  moment  of  inertia  about  any  other  a.\is  passing 
through  the  origin  and  having  cos  a  cos  /3  cos  7  for  its  direction- 
cosines. 

19.  If  the  centre  of  inertia  of  a  rigid  body  be  the  origin,  and 
the  principal  axes  at  that  point  the  axes  of  coordinates,  then  at 
an  umbilicus  of  the  ellipsoid 

1-2  1/2  r.2 


+  -.'^^^  + 


=  I, 


A^\     B+\     C+\ 

two  of  the  principal  moments  of  inertia  will  be  equal. 

20.  If  the  density  at  any  point  of  a  right  circular  cone  be 
proportional  to  the  distance  from  the  exterior  surface,  show  that 

the  radius  of  gyration  about  the  axis  of  figure  is  j-,  where  a  is 

the  radius  of  the  base. 

21.  Find  the  moment  of  inertia  of  the  solid 

(,l-2  +y  ^  .2  _  ,^^^.)2  =  ^2(  1-2  ^j,2  +  .2) 

about  the  axis  of  .v. 

Find  also  the  moment  of  inertia  of  the  surface  of  this  solid. 

22.  The  locus  of  points  at  which  one  of  the  principal  axes 
passes  through  a  fixed  point  in  one  of  the  principal  planes 
through  the  centre  of  inertia,  is  a  circle. 

23.  If  a  and  /'  be  the  sides  of  a  homogeneous  parallelogram, 
6  and  (/>  the  inclinations  of  its  principal  axes  in  its  own  plane, 


r  n 


I ' 


■  I 


iMJ 


So 


RIGID   DYNAMICS. 


through  its  centre  of  ineitia,  to  these  sides  respectively,  show 
^^^^  a^  sin  2  ^  =  />'^  sin  2  4>- 

24.  Find  the  moments  of  inertia  of  a  uniform  circular  lamina 
about  its  principal  axes  through  a  ^ivcii  point  in  its  plane. 

25.  Show  that  two  of  the  principal  moments  of  inertia  with 
respect  to  a  point  in  a  ri^id  body  cannot  be  equa".  unless  two 
are  equal  with  respect  to  the  centre  of  inertia  and  the  point  be 
situated  on  the  axis  of  unequal  moment. 

26.  Prove  that  in  any  rigid  body  the  locus  of  the  point 
through  which  one  of  the  principal  axes  is  in  a  given  direction 
is  a  rectangular  hyperbola  whose  plane  passes  through  the 
centre  of  inertia,  and  one  of  whose  asymptotes  is  in  the  given 
direction  ;  unless  the  given  direction  be  that  of  one  of  the  prin- 
cipal axes  through  the  centre  of  inertia. 

27.  A  series  of  parabolas  are  described  in  one  plane  having 
a  common  vertex  A  and  a  common  axis,  and  from  a  point  P  in 
one  of  them  an  ordinate  PJV  is  drawn  to  the  axis.  Show  that 
if  the  moment  of  inertia  of  the  curvilinear  area  APN  about  an 
axis  through  A  perpendicular  to  the  plane  of  the  parabolas  be 
proportional  to  the  area  APN,  the  locus  of  P  is  an  ellipse. 

28.  Show  that  if  the  momcntal  ellipsoid  at  a  point  not  in 
one  of  the  principal  planes  through  the  centre  of  inertia  be 
a  spheroid,  it  will  at  the  centre  of  inertia  be  a  sphere. 

29.  Find  the  moment  of  inertia  of  a  segment  of  a  circle 
about  its  chord. 

30.  Find  the  moment  of  inertia  of  an  equilateral  triangular 
lamina  about  an  axis  through  the  centre  of  inertia  and  perpen- 
dicular to  the  lamina  if  the  density  of  the  lamina  at  any  point 
varies  directly  as  the  distance  of  the  point  from  the  centre  of 
inertia. 

31.  If  A,  B,  C  be  the  moments  of  inertia  about  principal  axes 
through  the  centre  of  inertia  and  «,  ^,  7  be  the  moments  of 
inertia  about  principal  axes  through  a  point  P,  show  that 


MISCKLLANKOUS   EXAMPLES. 


i8l 


(I)  If  {a-^l3-r^)=A  +  B  -C,  the  locus  of  /'  will  be  one 
of  the  principal  planes  throu<;h  the  centre  of  inertia. 

(II)  If  rt  +  /t^  +  7  be  con.stant,  the  locus  of  P  will  be  a  sphere 
with  centre  at  centre  of  inertia. 

(III)  If  (  v«4- v^+ V7)(V^+  V7- V«) 

X  (  V7  +  V«  -  Vjg)(  \Ja  +  V^  -  V7) 

be  constant,  the  locus  of  P  will  be  an  ellipsoid  similar  and  simi- 
larly situated  and  concentric  with  the  central  ellipsoid  at  the 
centre  of  inertia. 

(IV)  If  ,8  —  y<a  and  P  lie  on  a  lemniscate  of  revolution 
having  for  foci  the  points  where  the  moniental  ellijjsoid  is  a 
sphere,  a  —  13  =  A  —  P,  a  and  l3  being  the  moments  about  the 
axes  through  P  which  pass  through  the  axis  A. 

32.  Find  the  moment  of  inertia  of  a  portion  of  the  arc  of  an 
equiangular  spiral  about  a  line  through  its  pole  perpendicular  to 
its  plane. 

33.  Find  the  moment  of  inertia  of  the  segment  of  a  ]:)arabolic 
area  bounded  by  a  chord  perpendicular  to  its  axis,  about  any 
line  in  its  plane  through  the  focus  ;  and  determine  the  position 
of  the  chord  that  all  such  moments  may  be  equal. 

34.  Prove  that  if  the  height  of  a  homogeneous  right  circular 
cylinder  be  to  its  diameter  as  V3  :  2,  the  moments  of  inertia  of 
the  cylinder  about  all  axes  passing  through  the  centre  of  inertia 
will  be  equal. 

35.  Find  the  moment  of  inertia  of  a  parabolic  area  bounded 
by  the  latus  rectum  about  the  line  joining  its  vertex  to  the 
extremity  of  its  latus  rectum. 

36.  Find  the  locus  of  those  diameters  of  an  ellipsoid,  the 
moments  of  inertia  about  which  are  equal  to  the  moment  of 
inertia  about  the  mean  axis. 

37.  One  extremity  of  a  string  is  attached  to  a  fixed  point;  the 
string  passes  round  a  rough  pulley  of  given  radius  and  over  a 


.    M 


I( 


lli 


1^  '■ 


182 


KI(;ilJ    DYNAMICS. 


smooth  peg  and  is  attached  to  a  weight  c(|iial  to  that  of  the 
jnillcy.  Determine  the  motion,  the  positions  of  the  string  on 
either  side  of  the  pulley  being  vertical. 

38.  A  heavy  uniform  rod  has  at  one  extremity  a  ring  which 
slides  on  a  smooth  vertical  axis ;  the  other  extremity  is  in 
motion  on  a  horizontal  plane,  and  is  connected  by  an  elastic 
string  with  the  point  where  the  axis  meets  the  plane.  Deter- 
mine the  motion  supposing  the  string  always  stretched. 

39.  A  ball  spinning  about  a  vertical  axis  moves  on  a  smooth 
horizontal  table  and  impinges  directly  on  a  perfectly  rough  ver- 
tical cushion.  Show  that  the  vis  viva  of  the  ball  is  diminished 
in  the  ratio  lo^^-f  i4tan2^:  lo-f  49tan2^,  where  f  is  the  co- 
effuient  of  restitution  of  the  ball  and  6  the  angle  of  reflection. 

40.  A  homogeneous  lamina  rotating  in  its  plane  about  its 
centre  of  inertia,  is  brought  suddenly  to  rest  by  sticking  a  two- 
pronged  fork  into  it.  Show  that  the  impulses  on  the  prongs  arc 
equal  to  one  another,  and  are  of  the  same  magnitude  wherever 
the  fork  is  stuck  in. 

41.  A  free  rod  is  at  rest  and  a  ball  is  fired  at  it  to  break  it. 
Show  that  it  will  be  most  likely  to  cause  it  to  break  if  it  strike 
it  at  the  midpoint,  or  at  one-sixth  of  its  length  from  either  end  ; 
and  that  it  will  be  least  likely  to  break  the  rod  if  it  strike  it  at 
one-third  of  its  length  from  either  end.  And  that  in  either  case 
the  most  likely  point  for  it  to  snap  is  the  middle  point. 

42.  Three  pieces  cut  from  the  same  uniform  rigid  wire  are 
connected  together  so  as  to  form  a  triangle  ABC,  which  is 
then  set  in  contact  with  a  smooth  horizontal  plane.  Find  the 
direction  and  magnitude  of  the  strains  at  the  angular  con- 
nections. 

Prove  the  following  construction  for  the  direction  of  the 
strains:  If  AB,  AC  he  produced  to  D,  E  respectively,  and 
BD  and  CE  be  each  made  equal  to  BC,  then  will  DE  be  paral- 
lel to  the  direction  of  the  strain  at  A. 


MISCKI.I.ANKOl'S    EXAMI'LKS. 


183 


Also  show  that  the  direction  of  the  strain  at  A  makes  with 


the  side  BC  an  an^^le 


=  tan' 


I     sin  /3  ■^  sin  ('    ) 
1  I  +  cos  13  +  cos  C) 


43.  The  ends  ol"  a  uniform  heavy  rod  move  on  the  same 
smooth,  fixed,  vertical  rin^^  Determine  its  anguhir  velocity  in 
the  lowest  position,  supposing  it  to  fall  from  a  given  position 
starting  from  rest. 

44.  A  body  revolves  about  a  horizontal  axis,  starting  from 
rest  when  the  centre  of  inertia  is  in  the  horizontal  plane  con- 
taining the  axis.  Show  that  when  the  body  has  revolved  through 
45°,  the  effective  force  u|)on  the  centre  of  inertia  makes  with 
the  vertical  an  angle  =  tan"'  3. 

45.  A  box  is  fixed  upon  a  horizontal  plane  and  its  lid  is 
placed  in  a  vertical  jiosition ;  a  blow  is  given  to  the  lid  at  the 
midpoint  of  the  upper  edge  and  jierpcndicular  to  its  plane. 
Determine  the  initial  imj^ulsc  on  the  hinges,  the  finite  pressure 
on  them  during  the  motion,  and  the  imj)ulsive  pressure  on  them 
when  the  lid  impinges  on  the  opposite  edge  and  closes  the  box. 

46.  AB,  BC  arc  two  equal  heavy  rods  hinged  together  at  B\ 
the  rod  AB  is  capable  of  moving  in  a  vertical  plane  about  A, 
and  C  can  slide  by  means  of  a  small  ring  along  a  vertical  axis 
passing  through  A.  Find  the  angular  velocity  with  which  the 
whole  must  revolve  about  AC  that  the  triangle  ABC  may  be 
equilateral. 

47.  A  string  with  one  end  fastened  to  a  smooth  vertical  wall 
is  wrapped  round  a  cylinder  which  is  then  placed  in  contact 
with  the  wall.  Find  the  velocity  of  the  cylinder  and  the  tension 
of  the  string  in  terms  of  the  inclination  rf  the  string  to  the  wall. 

48.  The  lower  extremity  ot  a  heavy  uniform  beam  of  length 
a  slides  on  a  weightless  incxtensible  string  of  length  2a,^  whose 
extremities  are  attached  to  two  fixed  points  in  a  horizontal  line, 
and  the  upper  extremity  slides  on  a  vertical  rod  which  bisects 
the  line  joining  the  fixed  points.     Prove  that  the  only  position 


1 84 


KIGIU   DYNAMICS. 


I" 


I  ■'   I 


I, 


of  equilibrium  of  the  beam  is  vertical  and  that  the  time  of  a 

-,  where 


2  TTtt 


small  oscillation   about  this   position   is    — 

2  y/{d^  —  l)-)  is  the  distance  between  the  two  fixed  points. 

49.  /'  pulls  Q  by  means  of  an  une.x'ensible  strinj;  passinpj 
over  a  rou^h  pulley  in  the  form  of  a  vesical  circle,  which  can 
turn  freely  about  an  a.xis  through  its  centre,  which  is  fixed. 
Determine  the  velocity  attained  a*'ter  a  given  space  has  been 
described. 

50.  A  hooj)  of  mass  ^f  rolls  down  a  rough  inclined  plane, 
and  carries  a  heavy  particle  of  mass  ;//  at  a  point  of  its  circum- 
ference.    Determine  the  motion. 

51.  A  hollow  tubular  ring  of  radius  n  contains  a  heavy 
])article  with  its  plain  vertical  ui)on  a  smooth  horizontal  plane ; 
a  horizontal  velocity  2^2ai^  is  communicated  to  the  ring  in  its 
own  plane.  Show  that  the  particle  will  just  rise  to  the  top  of 
the  tube. 

52.  Four  equal  particles  are  connected  by  four  equal  strings, 
which  form  a  square,  and  the  particles  repel  each  other  with  a 
force  varying  directly  as  the  distance.  If  one  of  the  strings  be 
cut,  find  the  velocity  of  each  particle  at  the  instant  when  they 
are  all  in  a  straight  line. 

53.  One  half  of  the  inner  surface  of  a  fixed  hemispherical 
bowl  is  smooth  and  the  other  half  rough  ;  a  solid  sphere  slides 
down  the  smooth  part  of  the  bowl,  starting  from  rest  at  the 
horizontal  rim,  and  at  the  bottom  comes  in  contact  with  and 
rolls  up  the  rough  part  of  the  surface.  Find  the  change  of  vis 
viva  of  the  sphere  at  the  bottom  of  the  bowl,  and  show  that  if 
6  be  the  angle  which  the  line  joining  the  centres  of  the  sphere 
and  bowl  makes  with  the  vertical  when  the  sphere  begins  to 
descend  the  rough  surface,  cos6?  =  |. 

54.  A  cone  of  mass  in  and  vertical  angle  2 «  can  move 
freely  about  its  axis  and  has  a  fine  smooth  groove  cut  along  its 
surface  so  as  to  make  a  constant  angle  B  with  the  generating 


i 


f 


c 
n 


MISCELLANEOUS   EXAMl'LES. 


l8i 


nc  of  a 

,  where 

nts. 

pcissing 
lich  can 
is  fixed, 
las  been 

d  plane, 
circum- 

a  lieavy 
il  plane ; 
ns  in  its 
le  top  of 

.1  strings, 
er  with  a 
trings  be 
len  they 

spherical 
ere  slides 
St  at  the 
with  and 
ige  of  vis 
ow  that  if 
he  sphere 
begins  to 

:an  move 
t  along  its 
renerating 


lines  of  the  cone.  A  heavy  particle  of  mass  /'  moves  .long 
the  groove  under  the  action  of  gravity,  the  system  being  initially 
at  rest  with  the  particle  at  a  distance  c  from  the  vertex.  Show 
that  if  0  be  the  angle  through  which  the  cone  has  turned  when 
the  particle  is  at  any  distance  ;-  from  the  vertex,  then 

;;//'2 +  /*/'- sin- «     ^ /i   •  .  ,j 

iHlr-\-I  r  snr  « 

k  being  the  radius  of  gyration  of  the  cone  about  its  axis. 

55.  A  heavy  ring  just  fitting  round  a  smooth  vertical  cylinder 
is  suspended  by  //  vertical  strings  of  ecpial  lengths,  and  fixed  to 
the  ring  at  eciuidistant  points.  When  an  angular  velocity  is  given 
to  the  ring  about  its  centre,  show  that  the  height  to  which  it 
rises  is  indejiendiMt  of  the  lengih  of  the  strings.  1^'ind  also 
the  greatest  value  of  the  angle  through  which  it  turns. 

56.  A  sphere  has  a  fine  wire  fastened  normally  to  a  point 
on  its  surface,  the  other  end  being  fastened  to  a  point  on  a 
rough  inclined  plane.  If  the  sphere  be  slightly  displaced  from 
its  position  of  equilibrium  on  the  plane,  find  the  time  of  a  small 
oscillation,  neglecting  the  weight  of  the  wire. 

57.  Two  equal  uniform  beams  /]/>,  AC  are  freely  movable  in 
a  vertical  plane  about  A,  />  and  C  are  connected  by  an  elastic 
string  whose  natural  length  is  ccpial  to  AB.  The  beams  are 
held  in  a  vertical  j)osition  and  suffered  to  descend.  Determine 
the  motion,  the  coefflcient  of  elasticity  of  the  string  being  equal 
to  four  times  the  weight  of  either  beam. 

58.  A  circular  wire  is  revolving  uniformly  about  its  centre 
fixed.  If  it  be  cracked  at  any  point,  show  that  the  tendency 
to  break  at  an  angular  distance  a  from  the  crack  is  proportional 

to  sin^ 


a 


59.  A  disc  which  has  a  particle  of  equal  mass  attached  to  its 
circumference,  rolls  on  a  rough  inclined  plane.  Determine  the 
motion  and  the  friction  in  any  position  of  the  disc,  supposing  it 


ill' 


186 


KKill)    IjVNAMICS. 


11. 


to  start  lioin  ilu-  |)()siti<)n  in  which   the   particle  is  in  contact 
with  the  |)l:inc. 

60.  A  spherical  shell  whose  centn-  is  fixed  contains  a  roii^^h 
ball  which  is  held  at  one  extremity  of  a  horizontal  diametc-r  ot 
the  shell  and  then  allowed  to  descend.  IT  the  radius  of  tl'  • 
shell  be  three  times  that  of  the  ball,  and  when  the  ball  is  next  in 
instantaneous  rest  the  same  point  of  each  is  a^ain  in  contact,  the 

anj;ular  velocity  of  the  line  joinin^^  their  centres  is  3\/ ]'''  [> 

0  beinj;  the  inclination  of  this  line  to  the  horizon,  and  (r  being 
till'  radius  of  the  ball. 

61.  A  circular  rinjjf  is  suspended  with  its  plane  horizontal,  by 
three  ecpial  vertical  inextensible  strings  attachi;d  at  ecpial  dis- 
tances to  its  circumference.  If  the  rinj;  be  twisted  till  the 
strin^^s  just  meet  in  a  point,  and  be  then  left  to  itself,  find  its 
ani^ular  velocity  when  the  strin<;s  are  vertical  aj^ain. 

62.  Two  rods  /i/>,  /)C  connected  by  a  hinj;e  at  B  are  in 
motion  on  a.  smooth  horizontal  plane,  the  end  A  beinj;  fixed. 
If  initially  AB  has  no  angular  velocity,  that  of  BC  being  oy,  show 

that  when  BC  has  no  angular  velocity,  that  of  AB  will  be  — 


and  the  ancrlc  between  the  rods  will  be  cos' 


3^ 


'  2n 


and  2  6  being  the  lengths  of  the  rods  which  are  supposed  equal 
in  mass. 

63.  A  uniform  heavy  beam  of  length  2r  is  supported  in  a 
horizontal  position  by  means  of  two  strings  without  weight,  each 
of  length  d,  which  are  fastened  to  its  ends,  the  other  ends  of  the 
strings  being  fixed ;  in  equilibrium  each  of  the  strings  makes  an 
angle  a  with  the  horizontal.  Find  the  time  of  a  small  oscillation 
when  the  system  is  slightly  displaced  in  the  vertical  plane  in 
which  it  is  situated,  the  strings  not  being  slackened. 

64.  A  lamina  bounded  by  a  cycloid  and  its  base  has  its  centre 
of  inertia  at  the  middle  point  of  its  axis.  It  is  placed  with  its 
base  vertical  on  a  perfectly  rough  horizontal  plane,  and  allowed 


contact 

a  rouj^h 

nictii'  <>t 

IS  of  the 

IS  lu-xt  ill 

itact,  the 
_i,'sin  ft 

.1  <i  hcuVfi; 

'.ontnl,  by 

qiKil  dis- 

:1  till   the 

If,  find  its 

/?  are  in 
in^  fixed. 
v^  (,),  show 

/ill  be  — 
2  a 

b      i 
osed  ecjual 

orted  in  a 
ei^ht,  each 
;nds  of  the 
;  makes  an 
oscillation 
d  plane  in 

s  its  centre 
ed  with  its 
nd  allowed 


V'SCKLLANKOUS   KXAMI'LKS. 


•«7 


to  roll  down.     Show  that  at  the  moment  its  vertex  reaches  the 

plane   its  aii-rnlar  velocity  is  %/  .,       ,.,        ,    where  a  is  the 

'  ■         \  I      ii-  -\-  A'      ] 

radius    ot    the  ^eneratin;;  circle  and   /•  llu-    radius  of   ;;yrati(»n 
about  the  centre  of  inertii. 

65.  A  wire  is  bent  into  the  form  of  the  lemniscate  i>^  =  (i^cos2ft, 
and  laid  upon  a  smooth  horizontal  table;  a  tly  walks  alon<^^  the 
top  of  the  wire,  starting;  from  one  vertex.  Show  that  if  the 
masses  of  the  wire  and  fly  be  in  the  ratio  d^://',  where  /(•  is 
the  radius  of  j^yration  of  the  lemniscate  about  a  vertical  axis 
throuL^h  the  node,  then   when  the  flv  has  arrived  at  the  node 


the  wire  has  turned  throuj;!!  an  aii<;le    ' 

43 

66.    A  uniform  circular  wire  of  radius  </,  movable  about  a  fixed 

point  in  its  circumference,  lies  on  a  smooth  horizontal  plane.     An 

insect  of  mass,  ecpial  to  that  of  the  wire,  crawls  alon^;  it,  startin-; 

from  the  extremity  of  the  diameter  opjiosite  to  the  fixed  ])oinl, 

its  velocity  relative  to  the  wire  bein^  uniform  and  ecpial  to  7'. 

Prove  that  after  a  time  /  the  wire  will  have  turned  throu<rh  an 


1     ''^         I    4.       1 
anf;le -  tan~' 

2  a 


I      ,  7'/ 

-  tan  — 


V3  Vv'3        2rt> 

6y.  A  uniform  strin<^  is  stretched  alonjjj  a  smooth  inclined 
l)lane  which  rests  on  a  smooth  horizontal  table,  l^noui^h  of  the 
strinf;  han^js  over  the  top  of  the  plane  to  keep  the  whole  system 
at  rest.  If  the  strint;  be  j^entlv  ])ulled  over  the  plane,  and  the 
whole  system  be  then  left  to  itself,  investigate  the  ensuing  motion, 
supposinjij  the  length  of  the  string  to  be  equal  to  the  height  of 
the  plane. 

68.  Two  particles  of  equal  mass  are  attached  to  the  extremi- 
ties of  a  rigid  rod  without  inertia,  movable  in  all  directions  about 
its  middle  point.  The  rod  being  set  in  motion  from  a  given 
position  with  given  velocity,  find  equations  to  determine  its  sub- 
uent  motion. 


scq 


md 


69.    A  rod  of  length  2  a  movable  about  its  lower  end  is  inclinec 
at  an  angle  a  to  the  vertical,  and  is  given  a  rotation  w  about  the 


1 


I  il' 


i  i 


•"■i  1 


!t 


i' 


■■^^<•■ 


!^l 


;i       8       ! 


if 


i88 


RIGID    DYNAMICS. 


vertical.     If  0  be  its  inclination  to  the  vertical  when  its  angular 
velocity  about  a  horizontal  axis  is  a  maximum,  show  that 

3  j^  sin^  0  tan  6  +  4  im^  sin*  a  =  o. 

70.  The  time  of  descent,  down  a  rough  inclined  plane,  of  a 
spherical  shell  which  contains  a  smooth  solid  sphere  of  the  same 
material  as  itself  is  ^j,  the  time  of  descent  down  the  same  plane 
of  a  solid  sphere  of  the  same  material  and  radius  as  the  shell  is 
/g.     Determine  the  thickness  of  the  shell. 

71.  A  heavy  chain,  flexible,  inextensible,  homogeneous,  and 
smooth,  hangs  over  a  small  pulley  at  the  common  vertex  of  two 
smooth  inclined  planes.  Apply  d'Alembert's  principle  to  deter- 
mine the  motion  of  the  chain. 

/2.  A  perfectly  rough  right  prism,  whose  section  is  a  square, 
is  placed  with  its  axis  horizonial  upon  a  board  of  equal  mass 
lying  on  a  smooth  horizontal  table.  A  vertical  plane  containing 
the  centres  of  inertia  of  the  two  is  perpendicular  to  the  axis  of 
the  prism ;  a  horizontal  blow  in  this  plane  communicates  motion 
to  the  system.  Show  that  the  prism  will  topple  over  if  the 
momentum  of  the  blow  be  greater  than  that  acquired  by  the 

I  3  TT 

system  falling  through  a  height  — tan  ~a,  where  «  is  a  side  of 

12  o 

the  square  section  of  the  prism. 

73.  Determine  the  small  or.ciliaiions  in  space  of  a  uniform 
heavy  rod  of  length  2  a,  suspended  from  a  fixed  point  by  an 
inextensible  string  of  length  /  fastened  to  one  extremity.  Prove 
that  if  X  be  one  of  the  horizontal  coordinates  of  that  extremity 
of  the  rod  to  which  the  string  is  fastened 

X  =  A  sin  {n^t  +  a)  +  B  sin  {n^t  +  ^), 

where  ;/j,  «2  are  the  two  positive  roots  of  the  equation 

aln^  -  (4^  +  3  l)g>fi  +  T^if  =  o 

and  A,  B,  a,  ^  are  arbitrary  constants. 


hL 


MISCELLANEOUS   EXAMPLES. 


189 


ts  angular 
lat 


lane,  of  a 
'  the  same 
ime  plane 
he  shell  is 

leous,  and 
tex  of  two 
le  to  deter- 


s  a  square, 
^qual  mass 
containing 
the  axis  of 
ates  motion 
over  if  the 
Ted  by  the 

is  a  side  of 


a  uniform 
)oint  by  an 
lity.  Prove 
.t  extremity 


74.  The  bore  of  a  gun-barrel  is  formed  by  the  motion  of  an 
ellipse  whose  centre  is  in  the  axis  of  the  barrel  and  plane  per- 
pendicular to  that  axis,  the  centre  moving  along  the  axis  and 
the  ellipse  revolving  in  its  own  plane  with  an  angular  velocity 
always  bearing  the  same  ratio  to  the  linear  velocity  of  its  centre. 
A  spheroidal  ball  fitting  the  barrel  is  fired  from  the  gun.  If 
V  be  the  velocity  with  which  the  ball  would  have  emerged 
from  the  barrel  had  there  been  no  twist,  prove  that  the  velocity 
of  rotation  with  which  it  actually  emerges  in  the  case  supposed  is 

2  77;/ 7' 


»n 


the  number  of  revolutions  of  the  ellipse  corresponding  to  the 
whole  length  /of  the  barrel  being  ;/,  and  i'  being  the  radius  of 
gyration  of  the  ball  about  the  axis  coinciding  with  the  axis  of 
the  barrel,  and  the  gun  being  supposed  to  be  immovable 

75  A  plane  lamina  moving  either  about  a  fixed  axis  or  in- 
stantaneously about  a  principal  axis,  impinges  on  a  free  inelastic 
particle  in  the  line  through  the  centre  of  inertia  of  the  lamina 
perpendicular  to  the  axis  of  rotation  at  the  moment  of  impact. 
If  the  velocity  of  the  particle  after  impact  be  the  maximum 
velocity,  prove  that  the  angular  velocity  of  the  lamina  will  be 
diminished  in  the  ratio  of  i  :  2. 

y6.  Two  equal  uniform  rods  are  placed  in  the  form  of  the 
letter  X  on  a  smooth  horizontal  plane,  the  upper  and  the  lower 
extremities  being  connected  by  equal  strings.  Show  that  which- 
ever string  be  cut  the  tension  of  the  other  will  be  the  same 
function  of  the  rods,  and  initially  is  |^''sin«,  where  a  is  the 
inclination  of  the  rods. 

yy.  An  equilateral  triangle  is  suspended  from  a  point  by 
three  strings,  each  equal  to  one  of  the  sides,  attached  to  its 
angular  points.  If  one  of  the  strings  be  cut,  show  that  the 
tensions  of  the  other  two  are  diminished  in  the  ratio  of  36  .  43. 


;  I 


¥ 


190 


RIGID   DYNAMICS. 


Iln: 


V. 


78.  Apply  t//r  principle  of  energy  to  determine  the  time  of  a 
small  oscillation  of  a  uniform  rod  placed  in  a  smooth,  f^ed,  hemi- 
spherical bowl,  the  motion  taking  place  in  a  vertical  plane. 

79.  A  frame  formed  of  four  equal  uniform  rods  loosely  jointed 
together  at  the  angular  points,  so  as  to  form  a  rhombus,  is  laid 
on  a  smooth  horizontal  plane  and  a  blow  is  applied  to  one  of  the 
rods  in  a  direction  at  right  angles  to  it.  Prove  that  the  frame 
will  begin  to  move  as  a  rigid  body  provided  the  middle  point  of 
the  rod  which  receives  the  blow  be  equidistant  from  the  line  of 
action  of  the  blow  and  the  perpendicular  dropped  upon  the  rod 
from  the  centre  of  inertia  of  the  frame. 

Prove  also  that  in  this  case  the  initial  angular  velocity  of  the 
rod  which  receives  the  blow  is  one-eighth  of  what  it  would  have 
been  had  it  been  unconnected  with  the  remaining  rods. 

80.  Three  equal  uniform  rods  AB,  EC,  CD,  freely  jointed  at 
B  and  C,  are  lying  in  one  straight  line  on  a  smooth  horizontal 
table,  and  an  impulse  is  applied  at  the  midpoint  of  BC,  perpen- 
dicular to  that  rod.  Find  the  stresses  on  the  hiT:ges  at  B  and  C 
in  any  subsequent  positions  of  the  rods,  and  show  that  when 
AB,  CD  are  perpendicular  to  BC,  their  midpoints  are  moving  in 
directions  which  make  an  angle  cos~^(^)  with  BC. 

81.  A  parallelogram  is  formed  of  four  rigid  uniform  rods 
freely  jointed  at  their  extremities.  If  the  parallelogram  be  laid 
on  a  smooth  horizontal  table  and  a  blow  be  applied  to  any  one 
of  the  rods  at  right  angles  to  it,  and  in  a  direction  passing 
through  the  intersection  of  the  lines  drawn  through  its  extremi- 
ties parallel  to  the  diagonals,  determine  the  init'al  motioi.  of  the 
parallelogram. 

82.  A  circular  disc  is  capable  of  motion  about  a  horizontal 
tangent  which  rotates  with  uniform  angular  velocity  w  about  a 
fixed  vertical  axis  through  the  point  of  contact.  Prove  that  if 
the  disc  be  inclined  at  a  constant  angle  a  to  the  horizontal, 


w'bma  =  -1-2. 


4.^ 
5«* 


MISCELLANEOUS   EXAMPLES. 


191 


c  of  a 
hcmi- 

ointcd 
is  laid 
of  the 
frame 

)oint  of 
line  of 

the  rod 


83.    A  uniform  rod  is  rotating  with  angular  velocity  a/ [^-M 

about  its  centre  of  inertia,  which  is  at  rest  at  the  instant  when 
the  rod,  being  vertical,  comes  in  contact  with  an  inelastic  plane 
inclined  to  the  horizontal  at  an  angle  sin~'\/i.  The  motion 
being  in  a  vertical  plane  normal  to  the  inclined  plane,  prove 
that  the  angular  velocity  of  the  rod  when  it  leaves  the  inclined 

plane  is  ^{4. 


V: 


r 


84.  If  a  rigid  body  which  is  initially  at  rest,  and  which  has  a 
point  in  it  fixed,  is  struck  by  a  given  impulsive  couple,  show  that 
the  vis  viva  generated  is  greater  than  that  which  would  have 
been  generated  by  the  same  couple  if  the  body  had  been  con- 
strained to  turn  about  an  axis  through  the  fixed  point  and  not 
coincident  with  the  axis  of  spontaneous  rotation. 

85.  A  and  B  are  two  fixed  points  in  the  same  horizontal  line; 
CD,  a  heavy  uniform  rod  equal  in  length  to  AB,  is  suspended  by 
four  inextensible  strings  AC,  AD,  BC,  BD,  where  /^Cis  equal  in 
length  to  BD,  and  AD  to  BC.  If  two  of  the  strings  AC,  BD  be 
cut,  determine  the  tension  oi  the  other  two  immediately  after 
cutting,  and  find  the  angular  velocity  of  the  rod  when  it  reaches 
its  lowest  position. 

86.  A  beam  AB  is  fixed  at  A.  At  B  is  fastened  an  elastic 
string  whose  natural  length  is  equal  to  AP\  the  other  end  of  the 
string  is  fastened  to  a  jjoint  C  vertically  above  A,  AC  being 
equal  to  AB.  The  beam  is  held  vertically  upwards  and  then 
displaced.  If  it  come  to  rest  when  hanging  vertically  down- 
wards, find  the  greatest  pressure  on  the  axis  during  the  motion. 

87.  Two  equal  rods  AB,  BC  are  connected  by  a  hinge  at  />. 
A  is  fixed  and  C  is  in  contact  with  a  smooth  horizontal  plane, 
the  system  being  capable  of  motion  in  a  vertical  plane.  If 
motion  commence  when  the  rods  arc  inclined  at  an  angle  «  to 
the  horizon,  show  that  there  will  be  no  pressure  at  the  hinge 
when  their  inclination  Q  is  given  by  the  equation 

3  (sin^  ^  +  sin  ^)=  2  sin  «. 


If-':! 


Ifl 


.11'  i 


1 


''  i 


liiS' 


It;   ■ 


l''f  >•, 


192 


RIGID   DYNAMICS. 


88.  A  rod  AB  is  movable  freely  in  a  vertical  plane  about  A  ; 
to  B  is  fastened  an  elastic  string,  the  other  end  being  attached 
to  a  point  C  in  the  vertical  plane  at  such  a  distance  from  A  that 
when  the  rod  is  held  horizontal  the  tension  on  the  string  vanishes. 
If  the  rod  be  now  allowed  to  fall,  find  the  modulus  of  elasticity 
of  the  string  that  the  rod  may  just  reach  a  vertical  position. 

89.  A  prolate  spheroid  is  fixed  at  one  of  its  poles,  and  is 
allowed  to  fall  from  its  position  of  unstable  equilibrium  under  the 
action  of  gravity  only.  Find  the  pressure  at  the  fixed  point  in 
any  subsequent  position. 

90.  Every  particle  of  two  equal  uniform  rods,  each  of  length 
2cT,  attracts  every  other  particle  according  to  the  law  of  gravita- 
tion ;  the  rods  are  initially  at  right  angles  and  are  free  to  move 
in  a  plane  about  their  midpoints,  which  are  also  their  centres  of 
inertia  and  are  coincident.  If  angular  velocities  w,  to'  be  com- 
municated to  the  rods  respectively,  show  that  at  the  time  /  the 
angle  6  between  them  is  given  by  the  equation 


y  =(.-«')H^iog 


(3-2V2)^i 


cos--|-sm  -+ 1 

2  2 


e ,  .  d 

cos--}-sm —  I 
2  2 


91.  Two  equal  spheres  of  radius  a  and  mass  M  slxq  attached 
to  the  extremities  of  a  rigid  rod  of  the  same  material,  whose 
length  is  4^-  and  section  -^^  of  a  principal  section  of  the  sphere. 
If  the  rod  can  move  freely  about  its  midpoint  and  one  sphere  be 
struck  by  a  blow  P  normal  to  it  and  the  rod,  the  time  which 
must  elapse  before  the  other  sphere  takes  the  place  of  this  one  is 

44  iraM 

yP 

92.  A  thin  uniform  rod,  one  end  of  w  hich  is  attached  to  a 
smooth  hinge,  ir  allowed  to  fall  from  a  horizontal  position. 
Prove  that  the  stress  on  the  hinge  in  any  given  direction  is  a 
maximum  when  the  rod  is  equally  inclined  to  this  direction 
und  to  the  vertical,  and  the  stress  perpendicular  to  this  is  then 


MISCELLANEOUS   EXAMPLES. 


193 


A; 
;hed 
that 
;hcs. 


Jg^/Fcosrt,  where  JFis  the  weight  of  the  rod  and  «  is  the  incli- 
nation of  the  given  direction  to  the  horizontal. 

93.  A  man  standing  in  a  swing  is  set  in  motion.  Supposing 
that  the  initial  inclination  of  the  swing  to  the  vertical  is  given, 
and  that  the  man  always  crouches  when  in  the  highest  position, 
and  stands  up  when  in  the  lowest,  find  how  much  the  arc  of 
vibration  will  be  increased  after  ;/  complete  oscillations. 

94.  Three  rods  are  hinged  together  so  as  to  form  an  isosceles 
triangle  ABC,  A  being  the  vertex.  The  whole  is  rotating  with 
angular  velocity  to  round  an  axis  through  the  middle  point  of 
the  base  and  perpendicular  to  the  plane  of  the  rods,  when  it  is 
suddenly  brought  to  rest.  Show  that  the  impulsive  action  at  A 
bisects  the  angle  BAC,  and  find  its  magnitude. 

95.  A  triangular  lamina  is  suspended  at  rest  horizontally  by 
vertical  strings  attached  to  its  angular  points  A,  B,  C.  If  the 
strings  at  B  and  C  be  simultaneously  cut,  show  that  there  will 
be  no  instantaneous  change  of  tension  in  the  string  at  A,  if  AD 
be  perpendicular  to  either  AB  or  AC.  AD  =  CD  cos  ADC, 
D  being  the  midpoint  of  BC. 

96.  A  hollow  spherical  shell  is  filled  with  homogeneous  fluid 
which  gradually  solidifies  without  alteration  of  density,  the 
solidification  proceeding  uniformly  from  the  outer  surface,  so 
that  the  mass  of  the  solidified  portion  is  proportional  to  the 
time.  If  the  shell  initially  rotate  about  a  given  axis  with  a 
given  angular  velocity  <u,  find  the  angular  velocity  at  any  subse- 
quent period  before  the  solidification  is  complete. 

97.  A  lamina  whose  centre  of  gravity  is  G  is  revolving 
about  a  horizontal  axis  perpendicular  to  it  and  meeting  it  in  C. 
Supposing  it  to  begin  to  move  from  a  position  in  which  CG  is 
horizontal,  prove  that  the  greatest  angle  which  the  direction 
of   the    pressure  on   the  axis  can   make  with    the  vertical    is 

cot"M — ^tan^  ,  where  6  is  the  correspondmg  angle  which  CG 

\3  ^i  ' 


*    f'l 


o 


'    s; 


f 


hit' 


M 


r  'i 


s^  ,j 


yi  I.- 


,!• 


^  '   i 


! 


f     •! 


194 


RICID   DYNAMICS. 


makes  with  the  vortical,  /•  is  the  radius  of  gyration  about  an  axis 
through  G  perpendicular  to  the  lamina,  and  //  =  CG. 

98.  A  rough  uniform  rod,  length  2  a,  is  placed  with  a  length 
r{>a)  projecting  over  the  edge  of  the  table.  Prove  that  the  rod 
will  begin  to  slide  over  the  edge  when  it  has  turned  through  an 


angle  tan"^ 


ixa^ 


a^  +  9{c-af 

99.  If  gravity  be  the  only  force  acting  on  a  body  capable  of 
freely  turning  about  a  fixed  axis  and  the  body  be  started  from 
its  position  of  stable  equilibrium  with  such  a  velocity  that  it 
may  just  reach  its  position  of  unstable  equilibrium,  find  the  time 
of  describing  any  angle. 

100.  If  an  isosceles  triangle  move,  under  the  action  of  gravity 
only,  about  its  base  as  a  fixed  axis  starting  from  a  horizontal 
position,  show  that  the  greatest  pressure  on  the  axis  is  seven- 
thirds  the  weight. 

loi.  If  the  centre  of  oscillation  of  a  triangle,  suspended  from 
an  angular  point  and  oscillating  with  its  plane  vertical,  lie  on 
the  side  opposite  the  point  of  suspension,  show  that  the  angle 
at  the  point  must  be  a  right  angle. 

102.  A  horizontal  circular  tube  of  small  section  and  given 
mass  is  freely  movable  about  a  vertical  axis  through  its  centre. 
A  heavy  particle  within  the  tube  is  projected  along  it  with  a 
given  velocity.  Given  the  coefficient  of  friction  between  the 
tube  and  the  particle,  determine  the  terminal  velocity  of  both, 
and  Lhe  time  which  must  elapse  before  that  motion  is  attained. 

103.  Part  of  a  heavy  chain  is  coiled  round  a  cylinder  freely 
movable  about  its  axis  of  figure  which  is  horizontal,  and  the 
remainder  hangs  vertically.  Determine  the  motion,  supposing 
the  system  to  start  from  rest  and  neglecting  the  thickness  of 
the  chain. 

104.  Two  weights  arc  connected  by  a  fine  chain  which  passes 
over  a  wheel  free  to  rotate  about  its  centre  in  a  vertical  plane. 


miscp:llani:ous  examples. 


195 


axis 


Given  the  coefficient  of  friction  between  the  string  and  the 
wheel,  find  the  condition  which  determines  whether  the  string 
will  slide  over  the  \\heel  or  will  not  slide. 

105.  Two  straight  equal  and  uniform  rods  are  connected  at 
their  ends  by  fine  strings  of  equal  length  a  so  as  to  form  a  par- 
allelogram. One  rod  is  supported  at  its  centre  by  a  fixed  axis 
about  which  it  can  turn  freely,  this  axis  being  perpendicular  to 
the  plane  of  motion,  which  is  vertical.  Show  that  the  middle 
point  of  the  lower  rod  will  oscillate  in  the  same  way  as  a  simple 
pendulum  of  length  a,  and  that  the  angular  motion  of  the  rods 
is  independent  of  this  oscillation. 

106.  A  loaded  cannon  is  suspended  from  a  fixed  horizontal 
axis,  and  rests  with  its  axis  horizontal  and  perpendicular  to  the 
fixed  axis,  the  supporting  ropes  being  equally  inclined  to  the 

vertical.     If  v  be  the  initial  velocity  of  the  ball  whose  mass  is  — 

of  the  weight  of  the  cannon,  and  //  the  distance  between  the 
axis  of  the  cannon  and  the  fixed  axis  of  support,  show  that 
when  the  cannon  is  fired  off  the  tension  of  each  rope  is  imme- 
diately changed  in  the  ratio  v^  -f  ii^gh  :  7i{n  -f  \)g/i. 

107.  Two  equal  triangles  ABC,  A'B'C,  right-angled  at  C 
and  C,  rotate  about  their  equal  sides  CA  and  A'C  as  fixed 
axes  in  the  same  horizontal  straight  line.  The  distance  CC 
is  less  than  the  sum  of  the  sides  CA,  A'C.  The  triangles, 
being  at  first  placed  horizontally,  impinge  on  one  another  when 
vertical.  Determine  the  initial  subsequent  motion  and  discuss 
the  case  in  which  A  A'  is  less  than  one-fifth  CC. 

:o8.  Find  the  envelope  of  all  the  axes  of  suspension  that  lie 
in  a  principal  plane  through  the  centre  of  inertia  of  a  rigid 
body,  and  such  that  the  length  of  the  simple  pendulum  may  be 
always  twice  the  radius  of  gyration  of  the  body  about  one  of 
the  axes  lying  in  the  plane. 

109.  A  flat  board  bounded  by  two  equal  parabolas  with  their 
axis  and  foci  coincident,  and  their  concavities  turned  towards 


196 


RIGID    DYNAMICS. 


■tn 


III 


;„ti' 


ii 


li '  i;i  ■' 


■■'i 

m 


each  other,  is  capable  of  niovinj;  about  the  tangent  at  one  of 
the  vertices.     Find  the  centre  of  percussion. 

1 10.  A  uniform  beam  capable  of  motion  about  its  middle 
point  is  in  equilibrium  in  a  horizontal  position ;  a  perfectly 
elastic  ball,  whose  mass  is  one-fourth  that  of  the  beam,  is 
dropped  upon  one  extremity  and  is  afterwards  struck  by  the 
other  extremity  of  the  beam.  Prove  that  the  height  from  which 
the  ball  was  dropped  was  |g(2«  +  i)7r  x  length  of  beam. 

111.  Two  equal  circular  discs  are  attached,  each  by  a  point 
in  its  circumference,  to  a  horizontal  axis,  one  of  them  in  the 
plane  of  the  axis  and  the  other  perpendicular  to  it,  and  each  is 
struck  by  a  horizontal  blow  which,  without  creating  any  shock 
on  the  axis,  makes  the  disc  revolve  through  90°.  Show  that 
the  two  blows  are  as  V6  :  V5. 

112.  A  rigid  body  capable  of  rotation  about  a  fixed  axis  is 
struck  by  a  blow  so  that  the  axis  sustains  no  impulse.  Prove 
that  the  axis  must  be  a  principal  axis  of  the  body  at  the  point 
where  it  is  met  by  the  perpendicular  let  fall  on  it  from  the 
point  of  application  of  the  blow. 

113.  A  uniform  rod  AB  of  mass  A/  is  freely  movable  about 
its  extremity  A,  which  is  fixed;  at  C,  a  point  such  that  AC  is 
horizontal  and  equal  to  AB,  a  smooth  peg  is  fixed  over  which 
passes  an  inelastic  string  fastened  to  the  rod  at  B,  and  to  a 
body  also  of  mass  M  which  is  supported  in  a  position  also 
below  C.  If  the  rod  be  allowed  to  fall  from  coincidence  with 
AC,  and  the  string  be  of  such  a  length  as  not  to  become  tight 
until  the  rod  is  vertical,  the  angular  velocity  of  the  rod  will  be 
suddenly  diminished  by  three-fifths. 

114.  A  piece  of  wire  is  bent  into  the  form  of  an  isosceles 
triangle  and  revolves  about  an  axis  through  its  vertex  perpen- 
dicular to  its  plane.  Find  the  centre  of  oscillation  and  show 
that  it  will  lie  in  the  base  when  the  triangle  is  equilateral. 


MISCELLANEOUS   EXAMPLES. 


197 


115.  A  circular  lamina  performs  small  oscillations 

(i)  about  a  tangent  line  at  a  given  point  of  its  circumference, 
(2)  about  a  line  through  the  same  point  perpendicular  to  its 
plane. 

Compare  the  times  of  oscillation. 

116.  A  uniform  beam  is  drawn  over  the  edge  of  a  rough  hori- 
zontal table  so  that  only  one-third  of  its  length  is  in  contact  with 
the  table ;  and  it  is  then  abandoned  to  the  action  of  gravity.  Show 
that  it  will  begin  to  slide  Ovcr  the  edge  of  the  table  when  it  has 

turned  through  an  angle  equal  to  tan"'  -,  ft  being  the  coefficient 
of  friction  between  the  beam  and  the  table. 

117.  A  uniform  beam  AB,  capable  of  motion  about  A,  is  in 
equilibrium.  Find  the  point  at  which  a  blow  must  be  applied 
in  order  that  the  impulse  at  A  may  be  one-eighth  of  the  blow. 

118.  A  rectangle  is  struck  by  an  impulse  perpendicular  to  its 
plane.  Determine  the  axis  about  which  it  will  begin  to  revolve, 
and  the  position  of  this  axis  with  reference  to  an  ellipse  inscribed 
in  the  rectangle. 

119.  A  rectangle  rotates  about  one  side  as  a  fixed  axis.  F'ind 
the  pressure  on  the  axis  (i)  when  horizontal,  (2)  when  inclined 
to  the  horizontal. 

120.  About  what  fixed  axis  will  a  given  ellipsoid  oscillate  in 
the  shortest  possible  time  ? 

121.  A  uniform  semicircular  lamina  rotates  about  a  fixed  hori- 
zontal axis  through  its  centre  in  its  plane.  Determine  the 
stresses  on  this  axis. 

122.  If  7"]  and  T^  are  the  times  of  a  small  oscillation  of  a 
rigid  body,  acted  on  only  by  gravity,  about  parallel  axes  which 
are  distant  a-^  and  a^  respectively  from  the  centre  of  inertia,  and 
T^be  the  time  of  a  small  oscillation  for  a  simple  pendulum  of 
length  rt-j  -f  a^y  then  will  {a^  —  ct^T"^  =  a^ T^  —  a^ T^. 

123.  A  uniform  beam  of  mass  ;;/,  capable  of  motion  about  its 
middle  point,  has  attached  to  its  extremities  by  strings,  each  of 


198 


RIGID   DYNAMICS. 


'i 


'V 


i 


!    I' 


IJ 

,1 

(if 


P1 


5Mi 


IcMf^th  /,  two  particles,  each  of  mass/,  which  haiif;  freely.  When 
the  beam  is  in  ec|uilibriiim,  inclined  at  an  an^le  «  to  the  vertical, 
one  of  the  strings  is  cut;  jjrove  that  the  initial  tension  of  the 

other  string  is  --.-iv-,  and  that  the  radius  of  curvature  of 

w  -\-  T,/>  sur «  .     .  „ 

.    9//siiv'« 

the  initial  ])ath  of  the  i)article  is  . 

'  ■  /;/  cos « 

124.  A  uniform  inelastic  beam  capable  of  revolving  about  its 
centre  of  inertia,  in  a  vertical  plane,  is  inclined  at  an  angle  a  to 
the  horizontal,  and  a  heavy  particle  is  let  fall  ujjon  it  from  a. 
point  in  the  horizontal  plane  through  the  upper  extremity  of  the 
beam.  Find  the  position  of  this  point  in  order  that  the  angular 
velocity  generated  may  be  a  maximum. 

125.  A  uniform  elliptic  board  swings  about  a  horizontal  axis 
at  right  angles  to  the  plane  of  the  board  and  passing  through 
one  focus.  Prove  that  if  the  cxcentricity  of  the  ellipse  be  V|, 
the  centre  of  oscillation  will  be  the  other  focus. 

126.  A  circular  ring  hangs  in  a  vertical  plane  on  two  pegs. 
If  one  peg  be  removed,  prove  that,  P^  P.,,  being  the  instanta- 
neous pressures  on  the  other  peg  calculated  on  the  supposition 
that  the  ring  is  (i)  smooth,  (2)  rough,  Pj^  ;  p^^  :  :  i  :  i  -|-|  tan^w, 
where  «  is  the  angle  which  the  line  drawn  from  the  centre  of 
the  ring  to  the  centre  of  the  peg  makes  with  the  vertical. 

127.  A  uniform  beam  can  rotate  about  a  horizontal  axis 
so  placed  that  a  ball  of  weight  equal  to  that  of  the  beam, 
resting  on  one  end  of  the  beam,  keeps  it  horizontal.  A  blow, 
perpendicular  to  the  length  of  the  beam,  is  struck  at  the  other 
end.  Investigate  the  action  between  the  ball  and  the  beam, 
and  the  stress  on  the  axis. 

128.  There  are  two  equal  rods  connected  by  a  smooth  joint; 
the  other  extremity  of  one  of  the  rods  can  move  about  a  fixed 
point,  and  that  of  the  second  along  a  smooth  horizontal  axis 
passing  through  the  fixed  point,  and  about  which  the  system  is 


MISCLLLANEOUS    KXAMl'LES. 


199 


v^. 


revolving  under  the  action  of  gravity.  Find  a  differential  e(|ua- 
tion  to  determine  the  inclination  of  the  rods  to  the  axis  at  any 
time. 

129.  An  elliptic  lamina  whose  e.xcentricity  is  J, Vio  is  sup- 
ported with  its  plane  vertical  and  transverse  axis  horizontal  by 
two  smooth,  weightless  pins  passing  through  its  foci.  If  one  of 
the  pins  be  suddenly  released,  show  that  the  pressure  on  the 
other  pin  will  be  initially  unaltered. 

130.  A  plane  lamina  in  the  form  of  a  circular  sector  whose 
angle  is  2 «,  is  .-n.spended  from  a  horizontal  axis  through  its 
centre,  perpendicular  to  its  plane.  Find  the  time  of  a  small 
oscillation,  and  show  that  if  3«  =4sin«  the  time  of  oscillation 
will  be  the  same  about  a  horizontal  axis  through  the  extremity 
of  the  radius  passing  through  the  centre  of  inertia  of  the  lamina. 

131.  A  hollow  cylinder  open  at  both  ends,  of  which  the  height 
is  to  the  radius  as  3  to  V2,  has  a  diameter  of  one  of  its  ends 
fixed.  Show  that  the  centres  of  percussion  lie  on  a  straight 
line  the  distance  of  which  from  the  fixed  axis  is  eight-ninths  of 
the  height  of  the  cylinder. 

132.  A  lamina  ABCD  is  movable  aoout  AB  as  a  fixed  axis. 
Show  that  if  CD  be  parallel  to  AB  and  AR^=t,  CD'^,  the  centre 
of  percussion  will  be  at  the  intersection  of  AC  wnd  BD. 

133.  In  the  case  of  a  rigid  body  freely  rotating  about  a  fixed 
axis,  show  that  in  order  that  a  centre  of  percussion  may  exist 
the  axis  must  be  a  principal  axis  with  respect  to  some  point  in 
its  length. 

134.  A  uniform  rod  movable  about  one  end,  moves  in  such 
a  manner  as  to  make  always  nearly  the  same  angle  a  with 
the  vertical.     Show  that  the  time  of  its  small  oscillations  is 


Vr       2(1  cos  a       ) 
Is.di+Scos'-^'Oi' 


a  being  the  length  of  the  rod. 


1 1 


i'l 


200 


KKJID   DYNAMICS. 


( 

|| 

>i 
.'I 


I 


1 1 .  tail  m 


r'  HI 


I 


4 


135.  One  end  of  :i  heavy  uniforn!  rod  slides  freely  on  a  fine 

smooth  wire  in  the  lorni  ol  an  ellipse  of  excentrieity  -  •',  and 

2 

axis  minor  equal  to  the  lenj^th  of  the  rod ;  the  other  end  of  the 
rod  slides  on  a  smooth  wire  coinciding  with  the  axis  minor  ot 
the  ellipse.  The  system  is  set  rotatinj^  about  the  latter  wire, 
which  is  fixed  in  a  vertical  position.  Prove  that  if  0  be  the 
inclination  of  the  rod  to  the  vertical  at  the  time  /,  u  the  initial 
value  of  0,  and  w  the  initial  an^adar  velocity  of  the  system  about 
the  vertical  axis,  cos  ^  =  cos  «  cos (w/  sin«), 

136.  A  lamina  in  the  form  of  an  ecpiilateral  trianj^le  rests 
with  its  base  on  a  horizontal  plane,  and  is  capable  of  moving;  iti 
a  vertical  plane  about  a  hin^^e  at  one  extremity  of  its  base. 
Prove  that   it  will  turn   completely  over  if   it  be  struck  at  its 

vertex  a  blow  greater  than  2mi-^(  '  ,  j  in  a  direction  perpen- 
dicular to  that  side  'vhich  docs  not  pass  through  the  hinge,  w 
being  the  mass,  </  the  length  of  a  side  of  the  lamina,  /•  its  radius 
of  gyration  about  an  axis  through  one  of  its  angular  points  per- 
pendicular to  its  plane. 

137.  In  the  case  of  the  motion  of  a  rigid  body  about  a  hori- 
2ontal  axis  under  the  action  of  gravity,  show  that  the  forces  are 
reducible  to  a  single  force  if  the  axis  be  a  principal  axis  at  the 
point  where  the  perpendicular  on  it  from  the  centre  of  gravity 
meets  it  and  not  otherwise.  If  the  horizontal  fixed  axis  be  a 
principal  axis  at  a  point  other  than  that  at  which  the  perpendic- 
ular on  it  from  the  centre  of  gravity  meets  it,  and  if  the  centre 
of  gravity  start  from  the  horizontal  plane  passing  through  the 
fixed  axis,  determine  the  pressures. 

138.  An  elliptic  paraboloid,  cut  off  by  a  plane  parallel  to  the 
tangent  plane  at  the  vertex,  is  capable  of  freely  rotating  about  a 
diameter  of  the  base  as  a  fixed  axis.  Find  the  line  of  action  of 
an  impulse  which,  acting  on  the  paraboloid,  produces  no  impulse 
on  the  fixed  diameter. 


MKSCICLLANKOUS    KXAMl'I.KS. 


201 


)n  a  fine 

.^3,  ami 

d  of  the 
minor  <tt 
tcr  wire, 
9  be  the 
he  initiiii 
cm  about 


i<;le  rests 

novin-;  in 

its  base. 

ick  at  its 

n  perpen- 

h'uVfi;c,  vt 

its  radius 

K)ints  per- 


out  a  hori- 
forces  are 
xis  at  the 
of  gravity 
axis  be  a 
perpendic- 
the  centre 
irough  the 

illel  to  the 
ng  about  a 

)f  action  of 
no  impulse 


i 


139.  If  a  rigid  body  have  a  centre  of  percussion  with  respect 
to  a  given  axis,  show  that  there  is  one  with  respect  to  any 
parallel  axis,  in  a  plane  containing  the  given  axis  and  the 
centre  of  inertia. 

140.  Investigate  the  angular  velocity  of  the  top  (l''ig.  51, 
page  1 17)  while  the  string  is  still  unwinding,  assuming  (  1  )  the 
tension  of  the  string  to  be  constant  and  the  axis  to  be  cylimhi- 
cal,  (2)  the  increase  of  the  tension  of  the  string  over  the  initial 
tension  to  vary  as  the  length  of  string  drawn  off  and  the  axis  to 
be  conical. 

141.  The  part  of  a  paraboloid  of  revolution  cut  off  by  a  plane 
through  the  focus  is  fixed  at  a  point  in  the  circumference  of  its 
circular  base.  If  it  be  .struck  by  a  blow  at  any  point,  in  a  direc- 
tion parallel  to  its  axis,  find  the  initial  instantaneous  axis. 

142.  If  but  one  force  act  on  a  rigid  body,  one  point  of  which 
is  fixed,  the  body's  angular  velocity  about  the  instantaneous  axis 
will  be  a  maximum  or  a  minimum  when  the  instantaneous  axis 
is  perpendicidar  to  the  direction  of  the  force. 

143.  A  uniform  rod  can  turn  freel}^  about  one  end  which  is 
fixed,  the  other  end  resting  on  a  smooth  inclined  plane.  If  it 
be  just  disturbed  from  its  position  of  unstable  ec|uilibriuni,  prove 
that  it  will  never  leave  the  plane  unless  its  inclination  to  the 
horizon  be  >  tan  ^{\  tan  B),  where  />'  is  the  semi-vertical  angle 
of  the  cone  described  by  the  rod. 

144.  A  rigid  body,  fixed  at  one  point  only,  is  in  motion  under 
the  action  of  finite  forces.  If,  throughout  the  motion,  the 
angular  acceleration  of  the  body  about  the  instantaneous  axis 
bear  to  the  moment  of  inertia  about  this  axis  and  to  the  forces 
acting  on  the  body  the  same  relation  as  if  the  axes  were  fixed, 
prove  that  if  the  three  principal  moments  of  inertia  at  the  fixed 
point  be  not  all  equal  the  locus  of  the  axis  relatively  to  the 
body  is  a  cone  of  the  second  order. 

145.  A  triangular  lamina  AJ^C  has  the  angular  point  C  fixed, 
and  is  capable  of  free  motion  about  it.     A  blow  is  struck  at  B, 


I 


If  i 
I'  I 


M  I 


RIGID   DYNAMICS. 

perpendicular  to  the  plane  of  the  lamina.  Show  that  the  instan- 
taneous axis  passes  through  one  of  the  points  of  trisection  of  the 
side  A  J). 

146.  Two  equal  uniform  rods  are  capable  of  motion  about  a 
common  extremity  which  is  fixed,  their  upper  ends  bei  \g  joined 
by  an  elastic  string.  They  are  set  in  vibration  about  a  vertical 
axis  bisecting  the  angle  between  them.  If  in  the  position  of 
steady  motion  the  natural  length  (2/)  of  the  string  be  doubled, 
the  modulus  of  elasticity  being  equal  to  the  weight  of  either 
rod,  then  the  angular  velocity  about  the  vertical  will  be 


nI{^€^1' 


where  /i  is  the  height  of  the  string  above  the  fixed  extremity. 

147.  A  rigid  body,  of  which  two  of  the  principal  moments  at 
the  centre  of  inertia  are  equal,  rotates  about  a  third  principal 
axis,  but  this  axis  is  constrained  to  describe  uniformly  a  fixed 
right  circular  cone  of  which  the  centre  of  inertia  is  the  vertex. 
Prove  that  the  resultant  angular  velocity  of  the  body  is  con- 
stant, that  the  requisite  constraining  couple  is  of  constant  mag- 
nitude, and  that  the  plane  or  the  couple  turns  uniformly  in  the 
body  about  the  axis  of  unequal  moment. 

148.  An  ellipsoid  is  rotating  with  its  centre  fixed  about  one 
of  its  principal  axes  (that  of  x)  and  receives  a  normal  blow  at  a 
point  {/i,  ky  I).  If  the  initial  axis  of  rotation  after  the  blow  lie 
in  the  principal  plane  oi  ya,  its  equation  is 

^2(^2  -f  r2)(rt2  _  b'^)iy  +  ^2(^2  +  ^2)(^2  _  ^2y.  _  q 

149.  A  sphere  whose  centre  is  fixed  has  an  elastic  string 
attached  to  one  point,  the  other  end  of  the  string  being  fastened 
to  a  fixed  point.  To  the  sphere  is  given  an  angular  velocity 
about  an  axis.  Give  the  equations  for  determining  it:  motion, 
the  string  being  supposed  stretched  and  no  part  of  it  in  contact 
with  the  surface  of  the  sphere.     If  the  natural  length  of  the 


A 


111 


!! 


i  ;' 


he  instan- 
ion  of  the 

1  about  a 
.  ig  joined 
a  vertical 
losition  of 
i  doubled, 
of  either 
ic 


:remity. 

loments  at 
I  principal 
nly  a  fixed 
the  vertex. 
»dy  is  con- 
istant  mag- 
mly  in  the 

about  one 
il  blow  at  a 
he  blow  lie 


astic  string 
ing  fastened 
dar  velocity 
it:  motion, 
it  in  contact 
ingth  of  the 


I 
I 


MISCELLANEOUS   EXAMPLES. 


203 


String  be  equal  to  a,  the  radius  o"  the  sphere,  and  it  be  fi.xed  at 
a  point  (9  at  a  distance  =  <'?(V2  —  i)  from  its  centre,  and  if  the 
sphere  be  turned  so  that  the  point  on  it  to  which  the  string  is 
fastened  may  be  at  the  opposite  extremity  of  the  diametor 
through  O,  prove  that  the  time  of  a  complete  revolution 


a 


^  V5  f'g- 


7r  + 


2V2 


where    ;,  ^  moclulus  of  elasticity  ^     UaM\ 
weight  of  sphere  \  vs  M  / 


where  fi  —  modulus  of  elasticity. 


3 


7r  + 


2V2 


150.  If  the  angular  velocities  of  a  rigid  body,  at  any  time  /, 
about  the  axes  a;  j',  c,  are  proportional  respectively  to 

cot(w  — ?/)/,    cot(;/  — /)/,    cot(/— w)/, 

determine  the  locus  of  the  instantaneous  axis. 

151.  A  uniform  rod  of  length  2  a  can  turn  freely  about  one 
extremity.  In  its  initial  position  it  makes  an  angle  of  90°  with 
the  vertical  and  is  projected  horizontally  with  an  angular  ve- 
locity ft).  Show  that  the  least  angle  it  makes  with  the  vertical 
is  given  by  the  equation  4  aco'^  cos  6  =  T,g  sin'^  6. 

152.  A  rigid  body  rotates  about  a  fixed  point  under  the 
action  of  no  forces.  Investigate  the  following  equations,  the 
invariable  line  being  taken  as  the  axis  of  ^ : 

-7-  =  —  6"  sin  ^  sin  (i  cos  <f)  f ) : 

dt  ^        ^\A     Br 

dyfr  _  ^fcos"^^      sin^(^\ 


defy 


d'yjr  _  G  cos^_ 


+  cos  e'^  = 

dt  dt  C 


G  denoting  the  angular  momentum  of  the  body,  and  the  other 
symbols  having  their  usual  meaning. 


i 


'11 


mi  i 


II  M 


ll'[,' 


i    1 
1.1 


■I  1}  < 


■}  .1 


I  ' 


204 


RIGID   DYNAMICS. 


153.  One  point  of  a  rigid  body  is  fixed  and  the  body  is  set  in 
motion  in  any  manner  and  left  to  itself  under  the  action  of  no 
force.  Prove  that  if  A,  B,  C  be  its  principal  moments  of  inertia 
at  the  fixed  point,  G  its  angular  momentum,  \  its  component 
angular  velocity  about  the  invariable  line,  w  its  whole  angular 
velocity,  the  component  angular  velocity  of  the  instantaneous 
axis  about  the  invariable  line  will  be 


X  + 


{G  -  AX)(G  -  BX)(G  -  C\) 


ABC(co'^  -  X^) 

154.  A  rod  is  fixed  at  one  end  to  a  point  in  a  horizontal 
plane  about  which  it  can  move  easily  in  any  direction.  When 
}*"  is  inclined  to  the  horizon  at  a  given  angle,  a  given  horizontal 
velocity  is  communicated  to  its  other  end.  What  will  hi  the 
velocity  and  direction  of  the  motion  of  the  free  end  at  the 
moment  when  the  rod  falls  on  the  horizontal  plane.? 

155.  AD,  EC  are  two  equal  rigid  rods  movable  about  a  pin 
at  L,  such  that  AL  —  DL  =  BC  =  CL,  and  their  ends  are  con- 
nected by  four  elastic  strings  of  equal  lengths.  If  the  beams 
are  made  to  revolve  in  opposite  directions  about  L  through  a 
given  angle,  and  then  the  system  be  left  to  itself,  determine  its 
subsequent  motion. 

156.  AB,  BC,  CD  are  three  equal  beams  connected  by  pins 
at  B  and  C  and  lying  in  the  same  right  line.  If  a  given  im- 
pulse be  communicated  to  BC  at  its  centre  in  a  direction  per- 
pendicular to  its  length,  determine  the  impulse  on  the  pins. 

157.  A  uniform  rod  is  free  to  rotate  about  its  extremity  in  a 
vertical  plane,  while  that  plane  is  constrained  to  revolve  uni- 
formly about  a  vertical  axis  through  the  extremity  of  the  rod. 
Show  that  if  the  rod  be  let  fall  from  an  inclination  of  30° 
above  the  horizon,  it  will  just  descend  to  the  vertical  position 
if  au)^  ■■=  3^,  where  <u  is  the  angular  velocity  of  the  plane  and  2  a 
is  the  length  of  the  rod.  Also  explain  the  nature  of  the  motion 
according  as  aa)^  is  less  than  3^^-  or  greater  than  3^'-. 


I 


MISCELLANEOUS   EXAMPLES. 


205 


is  set  in 
on  of  no 
)f  inertia 
mponent 

angular 
ntaneous 


lorizontal 
i.  When 
lorizontal 
ill  hi  the 
id  at  the 

out  a  pin 
3  are  con- 
he  beams 
through  a 
ermine  its 

;d  by  pins 
given  im- 
iction  pcr- 
;  pins. 

emity  in  a 
;volve  uni- 
)f  the  rod. 
ion  of  30° 
al  position 
ine  and  2  a 
the  motion 


I 


I 


158.  A  rigid  rod  of  given  mass  can  revolve  about  its  middle 
point  in  a  plane  inclined  at  a  given  angle  to  the  horizon.  A 
given  angular  velocity  is  communicated  to  both  rod  and  plane 
about  a  vertical  axis  through  the  middle  point  of  the  rod,  the 
sy.5tem  being  then  left  to  itself.  Show  that  the  rod  will  oscil- 
late about  its  horizontal  position. 

159.  One  of  the  principal  axes  of  a  body  revolves  uniformly 
in  a  fixed  plane,  while  the  body  rotates  uniformly  about  it. 
Determine  the  constraining  couple  and  show  that  if  the  mo- 
ments of  inertia  about  the  other  two  principal  axes  are  equal, 
the  couple  has  a  constant  moment. 

160.  A  body,  two  of  whose  principal  moments  are  equal,  is  free 
to  rotate  about  its  centre  of  gravity,  which  is  fixed  relatively  to 
the  earth's  surface.  Prove  that  if  the  body  be  made  to  rotate 
very  rapidly  about  its  principal  axis  of  unequal  moment,  that  axis 
will  move  both  in  altitude  and  azimuth,  and  that  if  the  motion 
in  altitude  be  prevented  and  the  axis  be  originally  placed  hori- 
zontally in  the  meridian,  it  will  be  in  a  position  of  equilibrium, 
stable  or  unstable,  according  as  the  rotation  is  from  west  to  east, 
or  from  east  to  west.  If  the  axis  be  originally  directed  in  any 
other  azimuth,  it  will  oscillate  about  its  position  of  stable  equi- 
librium nearly  in  the  same  way  as  the  simple  circular  pendulum 
whose  length  =  Bg/^AVlw  cosX),  where  A  and  B  are  the  princi- 
pal moments,  12  the  angular  velocity  of  the  earth  about  its  axis, 
0)  that  of  the  disc,  and  A,  the  latitude  of  the  place  of  experiment. 

161.  A  body  turning  about  a  fixed  point  of  it  is  acted  on  by 
forces  which  always  tend  to  produce  rotation  about  an  axis  at 
right  angles  to  the  instantaneous  axis.  Show  that  the  angular 
velocity  cannot  be  uniform  unless 

C-B  ^B-A  ^A-C     ^ 

A,  B,  C  being  the  principal  moments  of  inertia  with  respect  to 
the  fixed  point. 


'.     f'l 


206 


RIGID   DYNAMICS. 


162.  If  forces  act  on  a  homogeneous  spheroid  tencling 
always  to  produce  rotation  about  an  axis  «,  in  the  plane  of  the 
equator,  the  instantaneous  axis  will  describe  a  circular  cone  in 
the  body  abciut  its  polar  axis ;  but  the  angular  velocity  about 
the  instantaneous  axis  will  not  be  uniform  unless  the  axis  «  be 
always  at  right  angles  to  the  instantaneous  axis. 

163.  A  sphere  movalile  about  a  point  in  its  surface,  w^hich 
is  fixed  relatively  to  the  earth,  is  in  equilibrium  under  the  action 
of  gravity.  Suppose  the  earth  to  suddenly  cease  rotating  about 
its  axis,  find  the  instantaneous  axis  of  rotation  of  the  sphere 
and  show  that  the  angular  velocity  about  it  would  be 


CO  cos 


^^l,+(l+iii)%an^(,|, 


ft)  being  the  angular  velocity  of  the  earth,  fi  the  ratio  between 
its  radius  and  that  of  the  sphere,  and  6  the  latitude  of  the  place. 

164.  A  rigid  body  under  the  action  of  given  forces  is  in 
motion  about  a  fixed  point.  Defining  the  momental  plane  at 
any  instant  as  that  which  would  be  the  invariable  plane  if  the 
forces  affecting  the  body  were  at  that  instant  to  cease  acting, 
show  that  if  the  body  be  constantly  acted  upon  by  a  couple 
whose  plane  passes  through  the  instantaneous  .txis  and  is 
normal  to  the  momental  plane,  the  distance  of  the  momental 
plane  from  a  fixed  point  will  remain  unchanged.  If  the  body 
be  acted  upon  at  any  instant  by  an  impulsive  couple  in  the 
plane  referred  to,  show  that  the  tangent  of  the  angle  through 
which  the  momental  plane  is  suddenly  turned  varies  as  the 
moment  of  the  couple. 

'65.  A  body  is  moving  about  a  fixed  point  at  a  distance  P 
from  the  invariable  plane.  Assuming  that  the  central  ellipsoid 
rolls  upon  the  invariable  plane,  show  that  the  equation  to  the 
surface  generated  by  the  instantaneous  axis  in  the  body  is 

the  equation  to  the  ci-ntral  ellipsoid  being  A.x'^-\-Bj'^-{-C::^=^  i. 


< 


MISCELLANEOUS   EXAMPLES. 


207 


tending 
e  of  the 
cone  in 
;y  about 
ixis  «  be 

e,  which 
le  action 
ng  about 
e  sphere 


between 
he  place. 

ces  is  in 

plane  at 

me  if  the 

3e  acting, 

a  couple 

5    and    is 

m  omental 

the  body 

)le  in  the 

e  through 

es  as  the 

listancc  P 
d  ellipsoid 
ion  to  the 
ly  is 


166.  If  the  motion  of  a  rigid  body  about  a  fixed  point  in  it 
be  represented  by  three  coexistent  angular  velocities  o)^,  Wy,  w,, 
about  three  axes  mutually  at  right  angles,  show  that  all  the 

particles  in  a  cylindrical  surface  whose  axis  is  ^— = --  =  -'^  will 

rt)^      ft)^      w, 

have  linear  velocities  of  equil  magnitude.    (See  Prob.  2.  p.  iii.) 

167.  An  equilateral  triangular  lamina  is  revolving  in  its  own 
plane  about  its  centre  of  inertia.  If  one  of  the  angular  points 
becomes  suddenly  fixed,  show  that  the  lamina  will  rotate  about 
it  with  one-fifth  of  the  original  angular  velocity. 

168.  A  rigid  body  is  free  to  move  about  a  fixed  point,  and  in 
the  notation  of  Art.  62, 

Wj  =  ^?  sin  ^sin  ^,  (ii^  =  a€\ViO  qx)S<^,  (o^  =  a  cosO, 

find  the  position  of  the  body  at  any  given  time. 

169.  Show  from  Euler's  Equations  of  Motion  (Art.  60),  that 
when  no  impressed  forces  act,  no  axis  other  than  a  principal 
axis  can  be  a  permanent  axis. 

170.  When  a  body  is  acted  on  by  no  forces  and  moves  about 
a  fixed  point,  show  that  the  locus  of  the  instantaneous  axis  is  a 
conical  surface. 

171.  A  prolate  spheroid  of  revolution  is  fixed  at  its  focus;  a 
blow  is  given  it  at  the  extremity  of  the  axis  minor  in  a  line  tan- 
gent to  the  direction  perpendicular  to  the  axis  major.  Find  the 
axis  about  which  the  body  begins  to  rotate. 

172.  A  rigid  body  fixed  at  a  given  point  is  free  to  rotate  in 
any  way  about  that  point.  Given  the  angular  velocities  about 
three  axes  mutually  at  right  angles  and  fixed  in  space,  find  the 
velocity  of  any  point  in  the  body  and  the  vis  viva  of  the  whole 
system. 

173.  In  the  case  of  a  rigid  body  moving  about  a  fixed  point 
and  subject  to  the  action  of  no  forces  if  the  moment  {T  be  a  har- 


m 


m 


'?.* 


I! 


1"' 


II ': 


1,-1 
1 


i;:ii 


r  *  pi  I 


ii  *■ 


Ea     1 


I  ! 


i  ? 


l!  t  ,. 


'■(':     ., 


i 


208 


KIGID   DYNAMICS. 


monic  mean  between  the  moments  A  and  B,  and  the  instan- 
taneous axes  describe  the  separating  polhode,  then  will  (/>  be 
constant,  yjr  will  increase  uniformly,  and  tan  6  —  c"  tan  ^q,  where 


^=^<hi) 


174.  Intep^ratc  Euler's  equations  determininj;  the  motion  of  a 
rigid  body  about  a  fixed  point  for  the  case  in  which  no  forces 
act  and  two  of  the  principal  moments  are  equal. 

175.  If  a  body  be  in  motion  about  a  fixed  point  under  the 
action  of  no  external  forces,  show  that  the  angular  velocity 
about  the  radius  vector  of  the  momcntal  ellipsoid,  about  which 
the  body  is  turning,  varies  as  that  radius  vector,  and  that  the 
perpendicular  on  the  tangent  plane  at  the  extremity  of  the 
radius  vector  is  constant. 

176.  A  plane  lamina  of  uniform  density  and  thickness, 
bounded  by  a  curve  represented  by  the  equation  7'=a-{-bs,\vP-2d, 
moves  about  its  pole  as  a  fixed  point.  Show  that  if  the  lamina 
be  under  the  action  of  no  forces,  its  angular  velocity  will  be 
constant,  and  its  axis  will  describe  a  right  cone  in  space. 

177  A  lamina  in  the  form  of  a  quadrant  of  a  circle  is  fixed 
at  one  extremity  of  its  arc  and  is  struck  a  blow  perpendicular  to 
its  plane  at  the  other  extremity.  Find  the  velocities  generated 
and  the  pressures  on  the  fixed  point.  If  ^  be  the  inclination  of 
the  instantaneous  axis  to  the  radius  vector  through  the  fixed 
point,  show  that 

4.      Q       10—377 
tan^  = - 

iStt  — 10 

178.  The  point  (?  of  a  rigid  body  is  fixed  in  space,  but  the 
body  is  capable  of  free  motion  about  the  point.  OA,  OB,  OC 
are  the  principal  axes  and  A\  B',  C  are  the  principal  moments 
of  inertia  of  the  body  at  O.  Show  that  the  couple  necessary  to 
keep  the  body  moving  so  that  OC  shall  describe  a  cone  with 


i'l 


MISCELLANEOUS   EXAMPLES. 


209 


install- 
11  (f)  be 
,  where 


ion  of  a 
)  forces 

der  the 
velocity 
it  which 
:hat  the 
of  the 

ickness, 

(5'sin2  2^, 

2  lamina 

will  be 

!  is  fixed 
icular  to 
sncrated 
lation  of 
he  fixed 


but  the 

OB,  OC 

noments 

issary  to 

3ne  with 


semi-vertical  angle  «  uniformly  about  the  fixed  line  OZ  and  CO  A 
shall  maintain  a  constant  inclination  to  ZOC,  must  be  m  the 
plane  x{C  -  B')cos  /3  cos  u  +  y{C'  -  /i')sin  /3  cos  «  +  rj{A'  -  IV) 
sin/3sin«  =  o,  referred  to  OA,  OB,  OC  ^s  axes. 

179.  If  the  component  angular  velocities  of  a  rigid  body  about 
a  system  of  axes  fixed  in  space  be  ro^,  &>„,  &>„  and  those  about  a 
system  fixed  in  the  body  be  <Ui,  w,,,  rog,  and  if  these  coincide 
respectively  with  the  former  at  the  time  /,  prove  that 


d-(o.  d(ii,  .       dro,, 

-     -  —  wy — ?  +  ft),     -V. 

dt'^  (it         '  dt 

d\ 


Examine  this  and  get  the  equation  for         '  in  terms  of  w^,  &)^,  w^. 

180.  A  body  acted  on  by  no  forces,  and  having  one  point 
fixed,  is  such  that  if  A,  B,  C  are  the  principal  moments  of 
inertia  at  the  fixed  point,  (7  is  a  harmonic  mean  between  A  and 
B.  Show  that  if  6  be  the  angle  which  the  axis  of  C  makes  with 
the  invariable  line,  and  <^  the  angle  which  the  plane  of  CA 
makes  with  the  plane  through  the  invariable  line  and  the  axis 
of  C,  then  will  sin-  6  cos  2  0  be  constant. 

181.  A  rigid  lamina,  not  acted  on  by  any  forces,  has  one  point 
in  it  which  is  fixed,  but  about  which  it  can  turn  freely.  If  the 
lamina  be  set  in  motion  about  a  line  in  its  own  plane,  the 
moment  of  inertia  about  which  is  0,  show  that  the  ratio  of  its 
greatest  to  its  least  angular  velocity  is  A  -{-  Q  :  B  +  Q,  where 
A  and  B  are  the  principal  moments  of  inertia  about  axes  in  the 
plane  of  the  lamina.  If  the  lamina  in  the  previous  problem  be 
bounded  by  an  equiangular  spiral  and  the  intercept  of  the 
radius  vector  to  the  extremity  of  the  curve,  and  if  the  fixed 
point  be  the  pole, 

A  +  Q  :  B  ^-  Q  :  :  I  -\-  cos  2'y  siu^Cy  —  /3) :  i  —  cos  2 7 cos2('y  —  /3), 

where  the  extreme  radius  vector  is  inclined  to  one  principal 
axis  at  an  angle  7  and  to  the  initial  position  of  the  instantaneous 
axis  at  an  angle  /3. 


I 


t" 


I  I 


II' 


i;  1 


Li- 


;)! 


,r  W 


t;  i 


210 


RIGID   DYNAMICS. 


182.  A  smooth  ball  of  radius  a  moves  around  the  circum- 
ference of  a  disc  of  radius  r  +  ^r  and  of  four  times  the  mass  of 
the  ball;  the  disc  is  supjjorted  at  its  centre  and  provided  with  a 
rim  (whose  weight  may  be  neglected)  sufficient  to  keep  the  ball 
from  falling  off.  Show  that  the  velocity  of  the  ball,  in  order 
that  the  disc  may  maintain  a  constant  inclination  of  45°  to  the 
horizontal,  is 

2V2(r{r-{-(j)\ 


VI 


183.  A  rigid  lamina  in  the  form  of  a  loop  of  a  lemniscate 
r^  =  a^  cos  2  6,  not  acted  on  by  any  force,  is  started  with  a  given 
angular  velocity  about  one  of  the  tangent  lines  through  its  nodal 
point,  the  nodal  point  being  fixed.  Prove  that  its  greatest 
angular  velocity  has  to  its  least  angular  velocity  the  ratio 

V(37r  +  4):  V(37r). 

184.  A  rigid  body,  movable  about  a  fixed  point,  is  struck  a 
blow  of  given  magnitude  at  a  given  point.  If  the  angular 
velocity  thus  impressed  upon  the  body  be  the  greatest  possible, 
prove  that 


-7kk-wr°' 


where  A,  B,  C  are  the  moments  of  inertia  of  the  body  about  the 
principal  axes  at  the  fixed  point,  a,  b,  c  are  the  coordinates  of 
the  point  struck  in  relation  to  the  principal  axes  at  the  fixed 
point,  and  /,  m,  n  are  the  direction-cosines  of  the  line  of  action 
of  the  blow. 

185.  A  square  lamina  with  one  angle  attached  to  a  fixed  point 
rotates  about  a  side.  What  must  be  the  angular  velocity  of  the 
lamina  in  order  that  the  side  about  which  it  rotates  may  remain 
vertical .'' 

186.  A  rigid  body  is  rotating  about  an  axis  through  its  centre 
of  inertia,  when  a  certain  point  of  the  body  becomes  suddenly 
fixed,  the  axis  being  simultaneously  set  free.     Prove  that  if  the 


MISCELLANEOUS   EXAMPLES. 


211 


new  instantaneous  axis  be  parallel  to  the  orij^inal  fixed  axis,  the 
point  must  lie  in  the  line  represented  by  the  equations 


.r 


aVx  +  l)^my  +  c^mz  =  o,  {d^  -c^)'-t+  (c^  -  a^) ^+(a'^-d^)'^  =o, 

I  m  n 

the  principal  axes  through  the  centre  of  inertia  bein<;  taken  as 
axes  of  coordinates,  a,  b,  c  the  radii  of  gyration  about  these  lines, 
and  /,  1)1,  )i  the  direction-cosines  of  the  originally  fixed  axis 
referred  to  them. 

187  An  elliptic  lamina,  fixed  at  the  focus,  is  struck  in  a  direc- 
tion perpendicular  to  its  plane.  Find  the  instantaneous  axis 
and  show  that  if  the  blow  be  applied  at  any  point  of  the  ellipse 


y 


-^2 


the  angular  velocity  will  be  the  same,  the  focus  being  origin, 
and  the  axis  major  and  latus  rectum  the  axes  of  x  and  j  respec- 
tively, and  c  being  the  excentricity. 

188.  A  uniform  rod  of  length  a,  freely  movable  about  one  end, 
is  initially  projected  in  a  horizontal  plane  with  angular  velocity 
ft)  about  the  fixed  point.  If  6  be  the  angle  which  the  rod  makes 
with  the  vertical  and  0  be  the  angle  which  the  projection  of  the 
rod  on  the  horizontal  plane  makes  with  the  initial  position,  show 
that  the  equations  of  motion  are 

sin20#  =  a,,     rf-Y  =  ^-cos^-ft,2cot2^. 
(it        '     \dtJ       a 

Find  the  lowest  position  of  the  rod  and  if  this  be  when  6  =  —t 

3 
show  that  the  resolved  vertical  pressure  on  the  fixed  point  is 

then  equal  to  ^  of  the  weight  of  the  rod. 

189.  A  lamina  having  one  point  fixed  is  at  rest  and  is  struck 
a  blow  perpendicular  to  its  plane  at  a  point  whose  coordinates, 
referred  to  the  principal  axes  at  the  fixed  point,  are  a,  b.  Show 
that  the  equation  to  the  instantaneous  axis  is  a/P'x-\-bk'^}>  =  o, 
h,  k  being  the  radii  of  gyration  about  the  principal  axes.     Show 


uiSSiS 


212 


KIGIU   DVNAiMlCS. 


'I 


r,i':.! 


that  if  nh  lie  on  a  certain  straight  lino,  there  will  be  no  impulse 
on  the  fixed  point. 

190.  A  uniform  rod  of  lenjrth  2n  and  mass  ;;/,  capable  of  free 
rotation  about  one  end,  is  held  in  a  horizontal  position,  and  on 
it  is  placed  a  smooth  particle  of  mass  /  at  a  distance  c  from  the 

point,  r  being  <      ■;  the  rod  is  then  let  go.    Find  the  initial  pres- 
3 

sure  of  the  particle  on  the  rod,  and  sho-.,   that  the  radius  of 
curvature  of  the  particle's  path  is 

191.  A  lamina  in  the  form  of  a  symmetrical  portion  of  the 
curve  r=a{mr'^  —  6'^)  is  placed  on  a  smooth  plane  with  its  a.xis 
vertical,  then  infinitesimally  displaced  and  allowed  to  fall  in  its 
own  plane.  If  the  lamina  be  loaded  so  that  its  centre  of  inertia  is 
at  the  pole  and  its  radius  of  gyration  =  2  a,  find  the  time  in  which 
its  axis  will  fall  from  one  given  angular  position  to  another. 

192.  An  elliptical  lamina  stands  on  a  perfectly  rough  inclined 
plane.  Find  the  condition  that  its  equilibrium  may  be  stable, 
and  determine  the  time  of  a  small  oscillation. 

193.  A  perfectly  rough  plane,  inclined  at  a  fixed  angle  to  the 
vertical,  rotates  about  the  vertical  with  uniform  angular  velocity. 
Show  that  the  path  of  a  sphere  placed  upon  the  plane  is  given 
by  two  linear  differential  equations  of  the  form. 


dh' 


dfi 


.  dx  ,   „  d\x       .,dv  ,   „,        ^ 


the  origin  being  the  point  where  the  vertical  line,  about  which 
the  plane  revolves,  meets  the  plane ;  the  axis  of  y  being  the 
straight  line  in  the  plane  which  is  always  horizontal. 

194.  The  equal  uniform  beams  AB,  EC,  CD,  DE,  are  con- 
nected by  smooth  hinges  and  placed  at  rest  on  a  smooth  hori- 
zontal plane,  each  beam  at  right  angles  to  the  two  adjacent, 
so  as  to  form  a  figure  resembling  a  set  of  steps.     An  impulse 


MISCKLI.ANKOUS    KXAMIM.F.S. 


213 


I  pulse 


ing  the 


is  given  at  the  end  //,  along  .//)';  determine  the  impulsive  action 
on  any  hinge. 

195.  A  rectangle  is  formed  of  four  uniform  rods  of  lengths 
2(1  and  2/;  respectively,  which  are  connected  by  smooth  hinges 
at  their  ends.  The  rectangle  is  revolving  about  its  centre  on  a 
smooth  horizontal  plane  with  an  angular  velocity  o),  when  a 
point,  in  one  of  the  sides  of  length  2  a,  suddenly  becomes  fixed. 
Show  that  the  angular  velocity  of  the  sides  of  length  2  />  inimedi- 


ately  becomes 


jfo.     Find,  also,  the  change  in  the  angular 


6(1 +  4  d 
velocity  of  the  other  sides  and  the  impulse  at  the  point  which 

becomes  fixed. 

196.  A  uniform  revolving  rod,  the  centre  of  inertia  of  which 
is  initially  at  rest,  moves  in  a  plane  under  the  action  of  a  con- 
stant force  in  the  direction  of  its  length.  Prove  that  the  square 
of  the  radius  of  curvature  of  the  path  of  the  rod's  centre  of 
inertia  varies  as  the  versed  sine  of  the  angle  through  which  the 
rod  has  revolved  at  the  end  of  any  time  from  the  beginning  of 
the  motion. 

197.  Six  equal  uniform  rods  arc  freely  joined  together  and 
are  at  rest  in  the  form  of  a  regular  hexagon  on  a  smooth  hori- 
zontal plane.  One  of  the  rods  receives  an  impulse  at  its  mid- 
point, perpendicularly  to  its  length,  and  in  the  plane  of  the 
hexagon.  Prove  that  the  initial  velocity  of  the  rod  struck  is 
ten  times  that  of  the  rod  opposite  to  it. 

198.  A  uniform  rod  of  length  2  a  lies  on  a  rough  horizontal 
plane,  and  a  force  is  applied  to  it  in  that  plane  and  perpendicu- 
larly to  its  length  at  a  distance  7'  from  its  midpoint,  the  force 
being  the  smallest  that  will  move  the  rod.  Show  that  the  rod 
begins  to  turn  about  a  point  distant  -\/\(i^-\-/>'^)—p  from  the 
midpoint. 

199.  AB  is  a  rod  whose  end  A  is  fixed  and  which  has  an 
equal  rod  7)C  attached  at  B.  Initially  the  rods  AB,  BC  are  in 
the  same  straight  line,  AB  being  at  rest  and  BC  on  a  smooth 


'V'l  I 


!  I 


i 

!i 

;  I, 

i  ;i 
<  ;i 
I  ii 


214 


RI(;iI)    DYNAMICS. 


horizontal  plane  having  an  anj^ular  velocity  fo.  Show  that  the 
greatest  an^^le  between  the  rods  at  any  suhse{|uent  time  is 
cos"'  j''^  and  that  when  they  are  a^iiin  in  a  straight  line,  Ihcir 

angular  velocities  are  r^  and  —  -g-  respectively. 

200.  A  rectangular  board  moving  uniformly  without  rotation 
in  a  direction  parallel  to  one  side,  on  a  smooth  horizontal  plane, 
comes  in  contact  with  a  smooth  fixed  obstacle.  Determine  at 
what  point  the  impact  should  take  place  in  order  that  the 
angular  velocity  generated  may  be  a  minimum. 

201.  b'our  ecpial  uniform  rods,  freely  jointed  at  their  extremi- 
ties, are  lying  in  the  form  of  a  scpiare  on  a  smooth  horizontal 
table,  when  a  blow  is  applied  at  one  of  the  angles  in  a  direction 
bisecting  the  angle.  Find  the  initial  state  of  motion  of  each  rod, 
and  prove  that  during  the  subsequent  motion  the  angular  veloc- 
ity will  be  uniform. 

202.  A  sjihere  is  moving  at  a  given  moment  on  an  imperfectly 
rough  horizontal  table  with  a  velocity  ?-,  and  at  the  same  time 
has  an  angular  vtilocity  w  round  a  horizontal  diameter,  the  angle 
between  the  direction  of  v  and  the  axis  of  w  being  «.  Prove 
that  the  centre  of  the  sphere  will  describe  a  parabola  if 

rt'/'W  -f  (a^  —  P)vQ)  sin  <■  =  ai'^. 

203.  Two  rods,  OA  and  OB,  are  fixed  in  the  same  vertical 
plane,  with  the  point  O  upwards,  the  rods  being  at  the  same 
angle  «  to  the  vertical.  The  ends  of  a  rod  AB  of  length  2  a 
slide  on  them.  Show  that  if  the  centre  of  inertia  of  AB  be  its 
middle  point,  and  the  radius  of  gyration  about  it  be  /',  the  time 
of  a  complete  small  oscillation  is 


Mrt^tan^<  -f  X-^) 
\  i      (7 if  cot  a      i 


204.  One  end  of  a  heavy  rod  rests  on  a  horizontal  plane  and 
against  the  foot  of  a  vertical  wall ;  the  other  end  rests  against  a 
parallel  vertical  wall,  all  the  surfaces  being  smooth.  Show  that, 
if  the  rod  slijD  down,  the  angle  <}>,  through  which  it  will  turn 


tl 


h 
e( 
a: 


NflSCKI.I.ANI'OrS    KXAMI'I.KS. 


'5 


lat  the 
itnc  is 
I,  their 


[)tiiti()n 

pUmc, 

line  ut 

lat  the 

xtremi- 
rizonlal 
ircction 
ich  rod, 
r  vcloc- 

crfcctly 

nc  time 

ic  an<;lc 

Prove 


vertical 
"le  same 
igth  2  a 
B  be  its 
the  time 


iane  and 
igainst  a 
low  that, 
will  turn 


round  the  common  normal  to  the  vertical  walls,  will  he  j;iven  by 
thi'  eciualion  '—r-rl  i  +  "^cos'-A)-!-  ,  ..  ,.,  sin  d>  =  C,  where  2ii 
is  tlu  leiif^lh  of  the  rod  and  2  b  the  distance  betwecti  the  walls 

205.  Two  e(|ual  uniform  rods,  loosely  jointed  together,  are  at 
rest  in  one  line  on  a  smooth  horizontal  table,  when  one  of  them 
receives  a  horizontal  blow  at  a  {^iven  jjoint.  Determine  the  ini- 
tial circumstance  of  the  motion,  and  prove  that,  when  next  the 
rods  are  in  u  straight  line,  they  will  have  interchanged  angular 
velocities. 

206.  One  end  of  a  uniform  rod  of  wei;;ht  w  can  slide  by  a 
smooth  ring  on  a  vertical  rod,  the  other  entl  sliding  on  a  smooth 
horizontal  plane.  The  rod  descends  from  a  position  inclined  at 
an  angle  fi  to  the  horizon.  Show  that  the  rod  will  not  leave 
the  horizontal  i)lane  during  the  descent,  but  that  its  maximum 
pressure  against  it  is  J7ccos''*/i  and  that  its  ultimate  pressure 
is  \  w. 

207.  A  lamina  capable  of  free  rotation  about  a  given  point  in 
its  own  plane,  which  point  is  fixed  in  space,  moves  under  the 
action  of  given  forces.  If  the  initial  axis  of  rotation  of  the 
lamina  coincide  very  nearly  with  the  axis  of  greatest  moment 
of  inertia  in  the  plane  of  the  lamina,  the  angular  velocities  about 
the  other  principal  axes  will  be  in  a  constant  ratio  during  the 
motion. 

208.  A  sphere  of  radius  a  is  partly  rolling  and  partly  sliding 
on  a  rough  horizontal  plane.     Show  that  the  angle  the  direction 

of  friction  makes  with  the  axis  ot  .v  is  tan~' ,  //  and  ■:> 

V  —  (la).^ 

being  the  initial  velocities,  w^,  w,^  the  initial  angular  velocities. 

209.  A  perfectly  rough  circular  cylinder  is  fixed  with  its  axis 
horizontal.  A  sphere  is  placed  on  it  in  a  jiosition  of  unstable 
equilibrium,  and  projected  with  a  given  velocity  parallel  to  the 
axis  of  the  cylinder.     If  the  sphere  be  slightly  disturbed  in  a 


I' 


BHB 


ill  ! 


I  ^' 


.:1 


l;;M 


l*>i  t 


l»^[ 


111    '' 


2l6 


RIGID   DYNAMICS. 


horizontal  direction  perpendicular  to  the  direction  of  the  axis  of 
the  cylinder,  determine  at  what  point  it  will  leave  the  cylinder. 

2  i  D.  A  parabolic  lamina,  cut  off  by  a  chord  perpendicular  to 
its  axis,  is  kept  at  rest  in  a  horizontal  position  by  three  vertical 
strings  fastened  to  the  vertex  and  two  extremities  of  the  chord ; 
if  the  string  which  is  fastened  to  the  vertex  be  cut,  the  tension 
of  the  others  is  suddenly  decreased  one-half. 

211.  Three  equal,  perfectly  rough,  inelastic  spheres  are  in 
contact  on  a  horizontal  plane;  a  fourth  equal  sphere,  which  is 
rotating  about  its  vertical  diameter,  drops  from  a  given  height 
and  impinges  on  them  simultaneously.  Investigate  the  subse- 
quent motion. 

212.  A  rod  of  length  (7,  moving  w'th  a  velocity  v  perpendicu- 
lar to  its  length  on  a  smooth  horizontal  plane,  impinges  on  an 
inelastic  obstacle  at  a  distance  c  from  its  centre.      Show  that  its 


angular  velocity  when  the  end  quits  the  obstacle  is 


3^/c 


a' 


213.  A  solid  regular  tetrahedron  is  placed  with  one  edge  on 
a  smooth  horizontal  table  and  is  allowed  to  fall  from  its  position 
of  unstable  equilibrium.  Find  the  angular  velocity  of  the  tetra- 
hedron just  before  a  face  of  it  reaches  the  table,  and  the  magni- 
tude of  the  resultant  impulsive  blow. 

214.  A  uniform  sphere  of  radius  a,  when  placed  upon  two 
parallel,  imperfectly  rough,  horizontal  bars,  has  its  centre  at  a 
height  ^  above  the  horizontal  plane  which  contains  the  bars.  It 
is  started  with  a  velocity  7>  parallel  to  the  bars,  and  an  angular 
velocity  to  about  a  horizontal  axis  perpendicular  to  the  bars  in 
such  a  direction  as  to  be  diminished  by  friction.  In  the  case  in 
which  2a^D.  >  sdv,  the  sphere  will  begin  to  roll  after  a  time 

yw^(2 rtM-  5  ^^y 

where  fi  is  the  coefficient  of  friction.      What  will  at  that  instant 
be  the  velocity  and  position  of  the  sphere  ? 


3  axis  of 
ylindcr. 

licukir  to 
2  vertical 
ic  chord ; 
2  tension 

2S  are  in 

which  is 

m  height 

;he  subse- 

;rpendicu- 
ges  on  an 
)\v  that  its 

/c 

2    ■ 

le  edge  on 
ts  position 
[  the  tetra- 
the  magni- 

iipon  two 
:entre  at  a 
le  bars.  It 
an  angular 
:he  bars  in 
the  case  in 

a  time 


that  instant 


MISCELLANEOUS   EXAiMl'LES. 


217 


215.  A  heavy  uniform  rod  slips  down  with  its  extremities  in 
contact  with  a  smooth  horizontal  floor  and  a  smooth  vertical 
wall,  not  being  initially  in  a  plane  perpendicular  to  both  wall 
and  floor.  Prove  that  if  6  be  the  inclination  to  the  horizon  and 
(j)  the  angle  which  the  projection  of  the  rod  on  the  floor  makes 
with  the  normal  to  the  wall, 


(/1-2  +  ..2)sin(^^^'^"^^  ^,""^  ^^  =  /C-2cos0^'(^°^  ^  ^^"  ^\ 


and 


(/i'2  +  rt2)cos^sin<i^''^^-) 

dt^ 


=  k-  smO  — ^^ — 2v  _  (J  or  COS,  6  sm  6, 

dt^  ^ 

2  a  being  the  length  of  the  rod  and  k  its  radius  of  gyration 
about  an  axis  perpendicular  to  it  through  the  centre  of  inertia. 

216.  A  body  possesses  given  motions  of  translation  and  rota- 
tion referred  to  a  given  point  of  it.  Find  under  what  condition 
the  motion  may  be  exhibited  by  rotation  about  a  single  axis,  and 
the  equations  to  this  axis  when  the  condition  is  satisfied. 

217.  A  heavy  straight  rod  slides  freely  over  a  smooth  peg. 
Show  that  the  equations  to  its  motion  are 

dh'       (de\^ 


dt'' 


dt 


=  A'-sin0, 


and 


ii('-^+<}=..«os^, 


where  r  and  0  are  coordinates  of  the  centre  of  inertia  reckoned 
from  the  peg  and  a  horizontal  line. 

218.  A  smooth  wire  of  given  mass  is  bent  into  the  form  of  an 
ellipse  and  laid  upon  a  smooth  horizontal  table ;  an  insect  of 
given  weight  is  gently  laid  on  the  wire  and  crawls  along  it. 
Find  the  path  described  by  the  centre  of  the  elliptic  wire  and 
trace  it  on  the  table. 

219.  The  effect  of  an  earthquake  being  assumed  to  be  a  sud- 
den horizontal  displacement  in  a  given  direction  of  every  body 


i 


i; 

'lit  i 


re  1:1, 


i  ■V 


2l8 


KKllD    DYNAMICS. 


fixed  to  the  surface  of  the  earth,  explain  the  nature  of  the 
motion  caused  l)y  the  shock  in  the  half  of  a  uniform  cylindrical 
stone  column  which  is  cut  off  by  a  plane  bisecting  the  cylinder 
diagonally,  and  which  rests  with  its  base  ui)()n  a  fixed  horizontal 
plane,  friction  being  supposed  the  same  at  every  point. 

220.  If  a  rigid  body  initially  at  rest  be  acted  on  by  given  im- 
pulses, whose  resultant  is  a  single  impulse,  show  that  the  axis  of 
instantaneous  rotation  will  be  perpendicular  to  the  direction  of 
that  resultant. 

221.  A  circular  disc  rolls  down  a  rough  curve  in  a  vertical 
plane.  If  the  initial  and  final  positions  of  the  centre  of  the  disc 
be  given,  show  that  when  the  time  of  motion  is  the  least  pos- 
sible the  curve  on  which  the  disc  rolls  is  an  involute  of  a 
C)cloid. 

222.  A  circular  ring  is  free  to  move  on  a  smooth  horizontal 
plane  on  which  it  lies,  and  an  uniform  rod  has  its  extremities 
connected  with  and  movable  on  the  smooth  arc  of  the  ring. 
The  system  being  set  in  motion  on  the  plane,  show  that  the 
angular  velocity  of  the  rod  is  constant,  and  describe  the  paths 
of  the  centres  of  the  rod  and  ring. 

223.  A  wheel  whose  centre  of  gravity  docs  not  coincide  with 
the  centre  of  the  figure  is  allowed  to  roll  down  an  inclined 
plane  which  is  so  rough  as  to  prevent  sliding.  If  «  be  the  incli- 
nation of  the  plane,  a  the  radius  of  the  wheel,  //  the  distance  of 
its  centre  of  inertia  from  the  centre  of  the  figure,  and  /'  the 
radius  of  gyration  of  the  wheel  about  an  axis  through  its  centre 
of  inertia  perpendicular  to  its  plane,  show  that  when  the  wheel 
has  rolled  from  rest  through  an  angle  7,  the  resistance  exerted 
by  the  plane  either  equals  zero  or  is  normal  to  the  plane,  7 
being  given  by  the  equation, 

[tan«tan  .]  y\(n  +  hf  +  h''-«''\  +<f-f 

=  a* -  l((i -  // )2  +  B  +  a' l\(a  +  Iif  +  P - a^ \  tan  2«. 


of  the 
lulrical 
•ylinclcr 
rizontal 


ivcn  im- 
;ixis  of 
ction  of 

vertical 

the  disc 

:ast  pos- 

lue   of   a 

lorizontal 
:tremities 
the  ring, 
that  the 
the  paths 

icidc  with 
1  inclined 
5  the  incli- 
[istancc  of 
ind  /'  the 
its  centre 
the  wheel 
ce  exerted 
I  plane,  7 


MISCELLANEOUS   EXAMJ'LES. 


219 


;an  ^a. 


224.  A  sphere  on  a  smooth  horizontal  plane  is  placed  in  con- 
tact with  a  rough  vertical  plane  which  is  made  to  revolve  with  a 
uniform  angular  velocity  (o  about  a  vertical  axis  in  itself.  If  a 
be  the  initial  distance  of  the  point  of  contact  from  the  axis, 
r  the  distance  after  a  time  /,  and  c  the  radius  of  the  sjjhcre, 
prove  that  2r={a  +  c\/l)e'""^"+i(r-t:Vl)e-"'"^'t  Also  show  that 
as  /  increases  indefinitely,  the  ratio  of  the  friction  to  the 
pressure  approximates  to    1  :  V35. 

225.  A  free  plane  lamina  receives  a  single  blow  perpendicular 
to  its  plane.  Show  that  (i)  if  the  locus  of  points  where  the  blow 
may  have  been  applied  be  a  straight  line,  the  spontaneous  axis 
will  pass  through  a  determinate  point,  (ii)  if  the  locus  be  a 
circle  (centre  C),  the  spontaneous  axis  will  be  a  tangent  to  an 
ellipse  whose  axes  are  in  the  direction  of  the  principal  axes  at 
C  in  the  plane  of  the  lamina. 

226.  A  sphere,  in  contact  with  two  fixed  rough  planes,  rolls 
down  under  the  action  of  gravity.  If  2  a  be  the  angle  between 
the  planes  which  are  equally  inclined  to  the  horizon,  and  with 
v/hich  their  line  of  intersection  makes  an  angle  fS,  show  that  the 
acceleration  of  the  centre  of  the  spheres  is  uniform  and  equal  to 

5  sin'-^  a  sin  /9 


2  +  5  sin  a 


/r- 


227.  Three  equal  smooth  spheres  are  placed  in  contact,  each 
with  the  other  two,  on  a  smooth  horizontal  plane,  and  connected 
nt  the  points  of  contact.  A  fourth  equal  sphere  is  then  placed 
so  as  to  be  supported  by  the  other  three.  Supposing  the  con- 
nections between  the  three  spheres  suddenly  destroyed,  show 
that  the  pressure  between  the  fourth  sphere  and  each  of  the 
other  three  is  suddenly  diminished  by  one-seventh.  Also  deter- 
mine the  subsequent  motion. 

228.  A  sphere  is  placed  upon  two  smooth  equal  spheres  held 
in  contact,  and  these  rest  on  a  smooth  horizontal  plane  in  the 
position  of  equilibrium.     Show  if  the  spheres  be  left  to  them- 


i 


» 


m 


ir:  I 


220 


RIGID   DYNAMICS. 


selves,  the   pressure  on  the  upper  sphere   is   instantaneously 
diminished  to  six-sevenths  of  its  former  amount. 

229.  A  plane  lamina  lies  on  a  smooth  horizontal  table.  If  one 
point  of  it  be  constrained  to  move  uniformly  along  a  straight 
line  on  the  table,  show  that  the  lamina  will  revolve  about  the 
point  with  uniform  angular  velocity,  and  determine  the  magni- 
tude and  direction  of  the  force  of  constraint  at  any  time. 

230.  A  sphere  has  an  angular  velocity  about  a  horizontal 
diameter  and  falls  upon  a  rough,  inelastic  board  which  is  moving 
uniformly  in  a  horizontal  plane  in  the  direction  of  this  diameter. 
Find  the  initial  direction  of  the  motion  and  its  path  afterwards. 

231.  If  the  velocities  of  two  given  points  of  a  rigid  body  be 
given  in  magnitude  and  direction,  determine  the  velocity  of  any 
other  point  in  the  body. 

232.  Prove  that  any  motion  of  a  rigid  rod  may  be  represented 
by  a  single  rotation  about  any  one  of  an  infinite  number  of  axes, 
and  find  the  locus  of  these  axes. 

233.  A  free  ellipsoid  is  struck  a  blow  normal  to  its  surface. 
Show  that,  in  general,  there  is  no  axis  of  spontaneous  rotation. 

234.  A  free  rigid  body  is  at  a  certain  moment  in  a  state  of 
rotation  about  an  axis  through  its  centre  of  inertia,  when  another 
point  in  the  body  suddenly  becomes  fixed.  Prove  that  there 
are  three  directions  of  the  original  instantaneous  axis  for 
which  the  new  instantaneous  axis  will  be  parallel  to  it,  and  that 
these  directions  are  along  conjugate  diameters  of  the  momenta! 
ellipsoid  at  the  centre  of  inertia. 

235.  A  little  squirrel  clings  to  a  thin  rough  hoop,  of  which 
the  plane  is  vertical  and  is  rolling  along  a  perfectly  rough 
horizontal  plane.  The  squirrel  makes  a  point  of  keeping  a  con- 
stant altitude  above  the  horizontal  plane  and  selects  his  place 
on  the  hoop  so  as  to  travel  from  a  position  of  instantaneous  rest, 
the  greatest  possible  distance  in  a  given  time.     Prove  that  m 


MISCELLANEOUS   EXAMPLES. 


221 


eously 

If  one 
traight 
)ut  the 
magni- 

•izontal 
moving 
ameter. 
wards. 

)ody  be 
of  any 

esented 
of  axes, 

surface. 
)tation. 

state  of 
another 
at  there 
axis  for 
and  that 
lomental 

)f  which 
y  rough 
ig  a  con- 
his  place 
ious  rest, 
e  that ;;/ 


being  the  weight  of  the  squirrel  and  ;//  that  of  the  hoop,  the 
inclination  of  the  squirrel's  distance  from  the  centre  of  the  hoop 

to  the  vertical  is  equal  to  cos"^ 


m 


in  +  2  )n' 

236.  A  rough  homogeneous  sphere  rests  on  a  rough  horizon- 
tal plane ;  a  heavy  inelastic  beam  sliding  through  two  smooth 
rings  in  the  same  vertical  line  falls  upon  it  from  a  given  height. 
Find  the  position  of  the  sphere  relatively  to  the  beam,  in  order 
that  the  angular  velr";ity  communicated  to  the  sphere  maybe 
the  greatest  possible. 

237.  Two  equal  uniform  rods,  freely  jointed  tor^ether  at  one 
extremity  of  each,  are  at  rest  on  a  smooth  horizontal  plane. 
Find  the  point  at  which  either  must  be  struck  in  order  that  the 
system  may  begin  to  move  as  if  it  were  rigid. 

238.  A  heavy  beam  is  placed  with  one  end  on  a  smooth 
inclined  plane  and  is  left  to  the  action  of  gravity.  If  the  verti- 
cal plane  constraining  the  beam  be  perpendicular  to  the  inclined 
plane,  find  the  motion  of  the  beam  and  the  pressure  on  the 
plane  when  a  given  angle  has  been  turned  through. 

239.  A  disc  rolls  upon  a  straight  line  on  a  horizontal  plane, 
the  disc  moving  with  its  flat  surface  in  contact  with  the  plane. 

Show  that  the  disc  will  be  brought  to  rest  after  a  time  "^ 

where  v  is  the  initial  velocity  of  the  centre,  and  jx  the  coefficient 
of  friction  between  the  disc  and  the  table. 

240.  Determine  how  a  free  rigid  body  at  rest  must  be  struck 
in  order  that  it  may  rotate  about  a  fixed  axis. 

241.  A  uniform  bar  is  constrained  to  move  with  its  extremities 
on  two  fixed  rods  at  right  angles  to  each  other,  and  is  under  the 
action  of  an  attraction  varying  as  the  distance  from,  and  tending 
to,  the  point  of  intersection  of  the  rods.  Determine  the  time  of 
a  small  oscillation  when  the  bar  is  slightly  displaced  from  the 
position  of  rest. 


A 


i! 


\ 


K 


:    i 


Bmmam 


'fi'i 


M 


:,P 


I'  i'S  ' 


I  ' 


222 


RIGID   DYNAMICS. 


242.  A  heavy  cycloid,  the  radius  of  whose  generating;  circle 

is  -,  is  mounted  so  as  to  admit  of  sliding  in  a  vertical  plane 

4 
with  its  base  always  horizontal  and  so  that  every  point  of  it 

moves  in  a  straight  line,  inclined  at  an  angle  of  45°  to  the  hori- 
zontal. A  uniform,  smooth,  heavy  chaia  of  length  a  and  mass 
equal  to  that  of  the  cycloid  is  laid  over  it  so  as  to  be  in 
equilibrium  when  the  cycloid  is  supported ;  if  the  support  be 
suddenlv-  removed,  find  the  tension  at  any  point  at  the  com- 
mencement of  motion  and  show  that  it  is  a  maximum  at  a 
distance  from  the  vertex  given  by  the  equation 

8  TT  rt  =(96  -  7r2)^- -  32  V(rt2  -  ^-2). 

243.  A  body  is  turning  about  an  axis  through  its  centre  of 
inertia  ;  a  point  in  the  body  suddenly  becomes  fixed.  If  the  new 
instantaneous  axis  be  a  principal  axis  with  respect  to  the  point, 
show  that  the  locus  of  the  point  is  a  rectangular  hyperbola 

244.  A  uniform  rod  of  mass  w  and  length  2  a  has  attached 
to  it  a  particle  of  mass/  by  a  string  of  length  d.  The  rod  and 
string  are  placed  in  a  straight  line  on  a  smooth  horizontal  plane, 
and  the  particle  is  projected  with  velocity  z>  at  right  angles  to 
the  string.  Prove  that  the  greatest  angle  which  the  string  makes 
with  the  rod  is 


2  sm" 


12^ 


and  that  the  angular  velocity  at  the  instant  is 


7' 


a-\-b 


245.  A  rough  sphere  is  projected  on  a  rough  horizontal  plane 
and  moves  under  an  acceleration  tending  to  a  point  in  the  plane 
and  varying  as  the  distance  from  that  point.  Show  that  the 
centre  of  the  sphere  will  describe  an  ellipse,  and  find  its  com- 
ponent angular  velocities  in  terms  of  the  time. 

246.  Three  equal  uniform  rods,  AB,  BC,  CD,  freely  jointed 
together  at  B  and  C,  are  lying  in  a  straight  line  on  a  sraooth 


'j;  circle 
il  plane 

nt  of  it 
he  hori- 
id  mass 
0  be  in 
)port  be 
he  com- 
um  at  a 


;entrc  of 
the  new 
le  point, 
)ola 

attached 

rod  and 

:al  plane, 

angles  to 

ig  makes 


tal  plane 
he  plane 
that  the 
its  com- 

Y  jointed 
L  smooth 


MISCELLANEOUS   EXAMPLES. 


horizontal  plane  and  a  given  impulse  is  applied  at  the  midpoint 
of  BC  at  right  angles  to  BC.  Determim;  the  velocity  of  BC 
when  each  of  the  other  rods  makes  an  angle  0  with  it,  and 
prove  that  the  directions  of  the  stresses  at  B  and  C  make  with 
BC  angles  equal  to  tan"^(|  tan  0). 

247.  Three  equal  uniform  straight  lines,  AB,  BC,  CD,  freely 
jointed  together  at  B  and  C,  are  placed  in  a  straight  line  on  a 
smooth  horizontal  plane  and  one  of  the  outside  rods  receives  a 
given  impulse  in  a  direction  perpendicular  to  its  length  at  its 
midpoint.  Compare  the  subsequent  stresses  on  the  hinges 
with  the  impulse  given  to  the  rod. 

248.  A  homogeneous  right  circular  cylinder  of  radius  a, 
rotating  with  angular  velocity  w  about  its  axis,  is  placed  with  its 
axis  horizontal  on  a  rough  inclined  plane  so  that  its  rotation 
tends  to  move  it  up  the  plane.  If  u  be  the  inclination  of  the 
plane  to  the  horizontal  and  tan  «  the  coefficient  of  friction,  show 
that  the  axis  of  the  cylinder  will  remain  stationary  during  a 


period  T  = 


aw 


2  s:  sm  a 


and  that  its  angular  velocity  at  any  time  / 


during  this  period  is  equal  to  w  —  ."''' 


a 


249.  A  hoop  is  hung  upon  a  horizontal  cylinder  of  given 
radius.     Determine  the  time  of  a  small  oscillation 

I.    When  the  cylinder  is  rough. 
II.    When  the  cylinder  is  smooth. 

250.  Prove  the  following  equations  for  determining  the  mo- 
tion of  a  rigid  bod"  whose  principal  moments  of  inertia  "t  the 
Cjntre  of  inertia  are  equal : 


X     dn 


L      dw. 


—  =  -j-  vB^  +  TC'^o,  etc.,  -^  =  —1  -  ©./g  -f  0)3^2,  etc. ; 
Cj-       at  A        at 

It,  V,  10  being  the  velocities  of  the  centre  of  inertia  parallel  to 
the  three  axes  moving  in  space,  w^,  co.,,  tUg  the  angular  velocities 
about  these  axes,  6^  6,^,  6..  the  angular  velocities  of  these  axes 
about  fixed  axes  instantaneously  coincident  with  them,  X,  Y,  Z 


224 


RIGID   DYNAMICS. 


' .  »r 


1i  ^1    !    ' 


the  resolved  forces,  L,  M,  N  their  moments  about  the  axes, 
G  the  mass  of  the  body,  and  A  its  moment  of  inertia  about  any 
axis  through  the  centre  of  inertia. 

251.  A  uniform  rod  of  mass  ;//  and  length  2  a  has  attached 
to  it  a  particle  of  mass  /  by  means  of  a  string  of  length  /; 
the  rod  and  string  aie  placed  in  one  straight  line  on  a  smooth 
horizontal  plane,  and  the  pn.rticlc  is  projected  with  a  velocity 
V  at  right  angles  to  the  string.  Prove,  then,  when  the  rod 
and  string,  make  angles  B,  cf)  with  their  initial  positions, 


} 


^/e 


P  +  al?  cos(<^  -  6)  \  y  +\d^  +  ad  cos(<^ 


0) 


(it 


<fj— <^-);ff-KfJ=-- 


where 


3/  +  3  ^« 


252.  A  sphere  of  radius  a  is  projected  on  a  rough  horizontal 
plane  so  as  partly  to  roll  and  partly  to  slide.  If  the  initial 
velocity  of  translation  be  v,  the  initial  rotation  &>  about  a  hori- 
zontal axis,  and  the  direction  of  the  former  make  an  angle  a 
with  the  axis  of  the  latter,  show  that  the  angle  through  which 
the  direction  of  motion  of  the  centre  has  turned,  when  perfect 

rolling  begins,  is 

_i       2  aw  cos  a 


tan' 


$v  —  2  am  sin  a 


253.  If  a  homogeneous  sphere  roll  on  a  perfectly  rough  plane 
under  the  action  of  any  forces  whatever,  of  which  the  resultant 
passes  through  the  centre  of  the  sphere,  the  motion  of  the  centre 
of  inertia  will  be  the  same  as  if  the  plane  were  smooth  and  all 
the  forces  were  reduced  in  a  certain  constant  ratio  ;  and  the 
plane  is  the  only  surface  which  possesses  this  property. 

254.  A  smooth  ring  of  mass  ;;/  slides  on  a  uniform  rod  of 
mass  M.  Determine  the  velocity  of  the  ring  at  any  point  of  the 
rod  which  it  reaches,  no  impressed  forces  being  supposed  to  act. 


MISCELLANEOUS   EXAMl'LES. 


225 


L'  axes, 
)iit  Liny 

ttachcd 
nj^th  /; 
smooth 
velocity 
the  rod 


r  -t-  d)v, 


^rizontal 
e  initial 
;  a  hori- 
angle  a 
h  which 
1  perfect 


^h  plane 
resultant 
le  centre 
and  all 
and  the 

1  rod  of 
nt  of  the 
;d  to  act. 


If  when  the  ring  is  distant  c  from  the  centre  of  the  rod,  the 
angle  at  which  its  path  is  inclined  to  the  instantaneous  position 


of  the  rod  be  greater  than  cot"M  2  -f 


■{-' 


v/c- 


I 


,.,i »  show  that  it 

will  never  reach  the  centre  of  the  rod,  /c'^  being  the  radius  of 
gyration  of  the  rod  about  its  centre. 

255.  A  uniform  rod  of  weight  W^  and  length  2^  is  supported 
in  a  horizontal  position  by  two  rtne  vertical  threads,  each  of 
length  c,  and  each  is  attached  at  a  distance  c  from  the  centre 
of  the  rod.  The  rod  is  slightly  dis'  '  d  by  the  action  of  a 
horizontal  couple  whose  moment  is  ./K,  and  which  does  not 
move  the  centre  of  the  rod  out  of  a  vertical  line.  Show  that  the 
time  of  a  small  oscillation  of  the  rod  will  be 

256.  A  circular  lamina,  rotating  about  an  axis  through  the 
centre  perpendicular  to  its  plane,  is  placed  in  an  inclined  posi- 
tion on  a  smooth  horizontal  plane.  Give  a  general  explanation 
of  the  motion  deduced  from  dynamical  principles,  and  show  that 
under  certain  circumstances  the  la*  ^ina  will  never  fall  to  the 
ground,  but  that  its  centre  will  perform  vertical  oscillations,  the 
time  of  an  oscillation  bein'r 


TT 


/ 1  4-  4  cos^  a\h 


V 


2  U)' 


a 


sm  a 


a  being  the  inclination  of  the  lamina  to  the  horizon  at  first,  a 
its  radius,  and  to  its  angular  velocity. 

257.  A  beam  rests  with  one  end  on  a  smooth  horizontal  plane, 
and  has  the  other  suspended  from  a  point  above  the  plane  by  a 
weightless,  inextensible  string  ;  the  beam  is  slightly  displaced  in 
the  plane  of  beam  and  string.  Find  the  time  of  a  small  oscil- 
lation. 

Q 


1   * 


1 


III 


tr' 


226 


RICAD   DYNAMICS. 


I  u 


.11 


.11 


.iiif 


[/' 


ifll   1 


258.  Find  the  condition  that  :i  free  rigid  body  in  motion  may 
be  reduced  to  rest  by  a  sin<;le  blow. 

259.  A  perfectly  roii<;h  horizontal  plane  is  made  to  rotate 
with  constant  an<.;ular  velocity  about  a  vertical  axis  which  meets 
the  jjlane  in  0.  A  sphere  is  projected  on  the  plane  at  a  point 
P  no  that  the  centre  of  the  sphere  has  initially  the  same  velocity 
in  direction  and  magnitude  as  if  the  sphere  had  been  placed 
freely  on  the  plane  at  a  point  Q.  Show  that  the  sphere's  centre 
will  describe  a  circle  of  radius  OQ,  and  whose  centre  K  is  such 
that  O/'^  is  parallel  and  equal  to  OP. 

260.  If  a  free  rigid  body  be  struck  with  a  given  impulse,  and 
any  point  of  the  body  be  initially  at  rest  after  the  blow,  show 
that  a  line  of  points  will  also  be  at  rest,  and  determine  the  con- 
dition that  this  may  be  the  case  in  a  body  previously  at  rest. 

261.  A  free  rigid  body  of  mass  ///  is  at  rest,  its  moments  of 
inertia  about  the  principal  axes  through  its  centre  of  inertia 
being  A,  B,  C.  Supposing  the  body  to  be  struck  with  an  impulse 
R  through  its  centre  of  inertia,  and  with  an  impulsive  couple  G, 
prove  that  it  will  revolve  for  an  instant  about  an  axis  whose 
velocity  is  in  the  direction  of  its  length  and  equal  to 

LX  ^  MY    XZ 
Am      Bin      Cm 


A^    B'    cy 


X,  V,  Z  being  the  components  of  R,  and  Z,  M,  N  the  com- 
ponents of  G,  in  the  principal  planes. 

262.  A  sphere  with  a  sphere  within  it,  the  diameter  of  the 
latter  being  equal  to  the  radius  of  the  former,  is  placed  on  a 
perfectly  rough  inclined  plane,  with  the  centre  of  inertia  at  its 
shortest  distance  from  the  plane,  and  is  then  left  to  itself.  Find 
the  angular  velocity  of  the  body  when  it  has  rolled  round  just 
once,  and  determine  the  pressure  then  upon  the  plane. 


It 


IIP 


MISCKLLANKOUS    EXAMI'LKS. 


227 


)n  may 

rotate 

li  meets 

(a  point 

velocity 

placed 

f>  centre 

is  such 

Ise,  and 
vv,  show 
;hc  con- 
rest. 

lents  of 
•  inertia 
impulse 
3uple  G, 
s  whose 


he  com- 

•  of  the 
ed  on  a 
ia  at  its 
f.  Find 
md  just 


263.  Two  equal  rods  of  the  same  material  are  connected  by  a 
free  joint  and  placed  in  one  straij^ht  line  on  a  smooth  horizontal 
plane;  one  of  them  is  struck  perpendicularly  to  its  len<;th  at  its 
extremity  remote  from  the  other  rod.  Show  that  the  linear 
velocity  communicated  to  its  centre  of  inertia  is  one-fourth 
greater  than  that  which  would  have  been  communicated  to  it 
by  a  similar  blow  if  the  rod  had  been  free. 

In  the  subsequent  motion  show  that  the  minimum  angle 
which  the  rods  make  with  one  another  is  cos  '  /j. 

264.  AB,  BC,  CD  are  three  equal  uniform  rods  lying  in  a 
straight  line  on  a  smooth  horizontal  plane,  and  freely  jointed  at 
/)  and  C ;  a  blow  is  applied  at  the  midpoint  of  BC.  Show  that 
if  ft)  be  the  initial  angular  velocity  of  AB  or  CD,  6  the  angle 
which  they  make  with  BC  at  time  t, 

dO  ^  ft) 

dt      V^  I  +  sin'-^  e) 

265.  A  lamina  of  any  form  lying  on  a  smooth  horizontal 
plane  is  struck  a  horizontal  blow.  Determine  the  point  about 
which  it  will  begin  to  turn,  and  prove  that  if  c,  c^  be  the  dis- 
tances from  the  centre  of  inertia  of  the  lamina  of  this  point  and 
of  the  line  of  action  of  the  blow  respectively,  rr'  =/'^  where  k  is 
the  radius  of  gyration  of  the  lamina  about  the  vertical  line 
through  its  centre  of  inertia. 

266.  A  circular  lamina  whose  surface  is  rough  is  capable  of 
revolution  about  a  vertical  axis  through  its  centre  perpendicular 
to  its  plane,  and  a  particle  whose  mass  is  equal  to  that  of  the 
lamina  is  attached  to  the  axis  by  an  inelastic  string  and  rests  on 
the  lamina.  If  the  lamina  be  struck  a  blow  in  its  own  plane, 
determine  the  motion. 

267.  A  bicycle  whose  wheels  are  equal  and  body  horizontal 
is  proceeding  steadily  along  a  level  rough  road.  Obtain  equa- 
tions for  determining  the  instantaneous  impulses  on  the  machine 
when  the  front  wheel  is  suddenly  turned  through  a  horizontal 
angle  Q. 


1:  I    r 


}' 


(i    »J 


1(1' 


I' 


228 


RICH)    DYNAMICS. 


I  'i.i 


1    '1  ■  1  > 

^^H| 

^^1 

I 

EH 

|li|Hi 

^if  1 

Show  that  the  initial  hoii/oiital  angular  velocity  is  pro[)()r- 
tional  to  the  original  velocity. 

268.  The  radii  of  the  portions  of  a  horizontal  differential  axle 
of  weight  [Fare  ti  and  /',  and  their  lengths  are  /f  and  d.  The 
suspended  weight  is  also  [/'.  If  the  balancing  power  be  re- 
moved and  the  weight  be  allowed  to  fall,  show  that  in  time  it 
will  fall  through 


Or-/>f 


-A"^- 


3(rt  —  />f  +  2ab' 

269.  Show  how  to  determine  the  angular  velocities  of  a 
rotating  mass  by  observations  of  the  instantaneous  direction 
cosines  of  points  on  its  surface  referred  to  three  fixed  rectangu- 
lar axes,  and  their  time  rates  of  increase ;    i.e.  — ,  etc.     How 

dt 

many  such  observations  are  necessary  ? 

270.  A  sphere  composed  of  an  infinite  number  of  infinitely 
thin  concentric  shells  is  rotating  about  a  common  axis  under  no 
forces.  Assuming  that  the  friction  of  any  shell  on  the  consecu- 
tive external  one  at  any  point  varies  as  the  square  of  the  angular 
velocity  and  the  distance  of  the  point  from  the  axis,  obtain  the 

equation /v- — rw   ,   =  20)^   for  the  angular  velocity  at  any 

time  of  shell  of   radius  r,  and  show  that  the  solution  of  this 

equation  is  r^a  =/[  —  +  M'  where/  is  an  arbitrary  function. 

271.  An  Q,g^  with  its  axis  horizontal  is  rolling  steadily  round 
a  rough  vertical  cone  of  semi-vertical  angle  «.  The  shape, 
weight,  moment  of  inertia,  etc.,  of  the  egg  being  known,  find 
the  friction  acting,  and  the  time  of  completing  a  circuit. 

272.  A  vertical,  double,  elastic,  wire  helix  is  rigidly  attached  at 
one  end  to  a  horizontal  bar,  mass  J/,  and  is  constrained  io  retain 
the  same  radius  a.  When  in  equilibrium  the  tangent  angle  is  a. 
An  additional  weight  MgQ,  or  a  torsion  couple  MgaQ^  can  alter 
M  into  a-\-  6.     If  the  bar  be  depressed,  and  consequently  turned 


Misc'KLLANKOUS    KXAMl'I.KS. 


229 


ial  axle 
The 
he  re- 
time it 


through  ail  angle,  show  that  the  time  of  a  small  oscillation  will 

;    (cos  a  -f-      sin  a) .  ,  where  /  and  2  fid  are  the  lengths  ot 
'A''  3  ' 

the  helix  and  the  bar  respectively. 

273.  Four  equal,  smooth,  inelastic,  circular  discs  of  radius  d 
are  p'aced  in  one  plane  with  their  centres  at  the  four  corners 
of  a  square  of  which  each  side  =2  a.  They  attract  one  another 
with  a  force  varying  as  the  distance.  A  blow  is  given  to  one 
of  them  in  the  line  of  one  of  the  diagonals  of  the  square. 
Investigate  the  whole  of  the  subsequent  motion. 

274.  P  and  Q  are  two  points  in  a  uniform  rod  equidistant 
from  its  centre.  The  rod  can  move  freely  about  a  hinge  at  /'. 
The  hinge  is  constrained  to  move  up  and  down  in  a  vertical 
line.  If  the  motion  be  such  that  Q  moves  in  a  horizontal  line, 
determine  the  velocity  when  the  rod  has  any  given  inclination, 
the  rod  being  supposed  t    start  from  rest  in  a  horizontal  position. 

In  the  ca.se  in  which  the  whole  length  of  the  rod  =PQ^ I, 
show  that  the  time  of  a  comjilete  oscillation  is  (2  7r)^(r  |)"'f 

275.  A  circular  and  a  semicircular  lamina  of  equal  radii  a 
are  made  of  the  same  material,  which  is  perfectly  rough.  Their 
centres  are  joined  by  a  tight  inelastic  cord;  also  the  centre  of 
the  circular  lamina  is  joined  to  the  highest  point  of  the  semi- 
circular lamina  by  a  string  of  length  ^j:V3.  The  semicircular 
lamina  stands  with  its  base  on  a  perfectly  rough,  inelastic  plane. 
The  circular  lamina  rests  on  the  top  of  the  semicircular  lamina 
and  in  the  same  vertical  plane  with  it.  It  is  disturbed  from  its 
position  of  equilibrium.     Prove  that  just  after  it  has  struck  the 

plane  its  angular  velocity  =  —  \\]- 

276.  A  uniform  rod,  capable  of  free  motion  about  one  extrem- 
ity, has  a  particle  attached  to  it  at  the  other  extremity  by  means 
of  a  string  of  length  /  and  the  system  is  abandoned  freely  to 
the  action  of  gravity  when  the  rod  is  inclined  at  an  angle  «  to 


!30 


RIGID   DYNAMICS. 


the  horizon  and  the  strinir  is  vertical.     Prove  that  the  radius  of 


i  u       I 


y        I 


il: 


't-5 


curvature  of  the  particle's  initial  path  is 


9/ 


)n-\-2p 


cos''^  a 


m         sin«(2  — 3sin-«)' 
m  and/  being  the  masses  of  ti.e  rod  and  particle  respectively. 

277.  To  a  smooth  horizontal  plane  is  fastened  a  hoop  of 
radius  r,  which  is  rough  inside,  /u,  being  the  coefficient  of  friction. 
In  contact  with  this  a  disc  of  radius  a  is  spun  with  initial  angu- 
lar velocity  ;/  and  its  centre  is  projected  with  velocity  v  in  such 

a  direction  as  to  be  most  retarded  by  friction.     Show  that  after 

an  +  V 


a  time 


c~a      B 


{P"!' 


ajiB 


the  disc  will  roll  on  the  inside  of 


the  hoop. 

278.  An  elephant  rolls  a  homogeneous  sphere  of  diametf^r  a 
inches  and  mass  ^  directly  up  a  perfectly  rough  plane  inclined 
/3  to  the  horizon,  by  balancing  himself  at  a  point  disiant  «  from 
the  sphere's  highest  point  at  each  instant.  Show  that,  the 
elephant  being  conceived  as  without  magnitude  but  of  mass  E, 
he  will  move  the  sphere  through  a  space 

fi     ,^     E  &ma  —  {E-\-  S)&\n  ^ 

2      a      ii  cos(«-}-/^)  +  ii+-|-5* 
where  t  is  the  time  elapsed  since  the  commencement  of  the 
motion. 

279.  A  circular  disc  of  mass  M  and  radius  r  can  move  about 
a  fixed  point  A  in  its  circumference,  and  an  endless  fine  string 
is  wound  round  it  carrying  a  particle  of  mass  m,  which  is  initially 
projected  from  the  disc  at  the  other  end  of  the  diameter  through 
A,  with  a  velocity  v  normally  to  the  disc,  which  is  then  at  rest. 
Show  that  the  angular  velocity  of  the  string  will  vanish  when 
the  length  of  the  string  unwound  is  that  which  initially  sub- 
tended at  the  point  A  an  angle  /3  given  by  the  equation 

( yS  tan /8  +  I )  cos'"^  yS -f  ^— =  o, 

8  ;// 

and  that  the  angular  velocity  of  tlie  disc  is  then  -'—(2  -f-/3  tan  /9)~'. 


MISCELLANEOUS   EXAMPLES. 


231 


radius  of 


ctively. 

hoop  of 
:  friction. 
;ial  angu- 
i'  in  such 
hat  after 

inside  of 


ameter  a 
inclined 
it  a  from 
that,  the 
■  mass  E, 


It  of  the 


wo  about 
ne  string 
s  initially 
•  through 
n  at  rest, 
ish  when 
ially  sub- 
1 


ftan/8)-i. 


280.  A  uniform  rod  AB  can  turn  freely  about  the  end  A, 
which  is  fixed,  the  end  B  being  attached  to  the  point  C,  distant  c 
vertically  above  A,  by  an  elastic  string  which  would  be  stretched 
to  double  its  length  by  a  tension  equal  to  the  weight  of  the  rod. 
If  the  rod  be  in  equilibrium  when  horizontal  and  be  slightly 
displaced  in  a  vertical  plane,  prove  that  the  period  of  its  small 

oscillations  is v("^)»  where  p  is  the  stretched  length  of  the 

string  in  equilibrium, 

281.  A  hollow  cylinder,  of  which  the  exterior  and  interior 
radii  are  a  and  b,  is  perfectly  rough  inside  and  outside,  and  has 
inside  it  a  rough  solid  cylinder  of  radius  c.  When  the  two  are 
in  motion  on  a   perfectly  rough  horizontal  table,   prove   that 

where  M  and  m  are  the  masses  of  the  hollow  and  the  solid 
cylinder  respectively,  ^  the  angle  the  hollow  cylinder  has 
turned  through,  and  0  the  angle  which  the  plane  containing 
their  axes  makes  with  the  vertical  after  the  time  t. 

282.  A  string  of  length  Cy  fixed  at  one  end,  is  tied  to  a  uniform 
lamina  at  a  point  distant  b  from  the  centre  of  inertia.  The 
centre  of  inertia  is  initially  at  the  greatest  possible  distance 
from  the  fixed  point  and  has  a  velocity  v  given  to  it  in  the  plane 
of  the  lamina  and  perpendicular  to  the  string.  Prove  that  when 
the  angle  between  the  string  and  line  ^  is  a  maximum,  the 

angular  velocity  of  the  lamina  is   - —  and  the  tension  of  the 


string  is 


2vnh      K''--2c{b  +  c)         ""^^ 


,  J/^2  being  the  greatest  moment 


{b^cf     K'-4c{b  +  c) 
of  inertia  of  the  lamina  at  the  centre  of  inertia. 

283.  In  a  circular  lamina  which  rests  on  a  smooth  horizontal 
table  and  which  can  turn  freely  about  its  centre,  which  is  fixed, 
a  circular  groove  is  cut.  If  a  heavy  particle  be  projected  along 
the  groove,  supposed  rough,  with  given  velocity,  find  the  time 
in  wnich  the  particle  will  make  a  complete  revolution  (i)  in 
space,  (ii)  relatively  in  the  groove. 


232 


RIGID   DYNAMICS. 


It 


hi 


284.  Four  equal  rods,  each  of  mass  m  and  length  /,  are  con- 
nected by  smooth  joints  at  their  extremities  so  as  to  form  a 
rhombus.  A  constant  force  mf  is  applied  to  each  rod  at  its 
middle  point,  and  perpendicular  to  its  length,  —  each  force  tend- 
ing outwards.  If  the  equilibrium  of  the  system  be  slightly 
disturbed  by  pressing  two  opposite  corners  towards  each  other, 
and  the  system  be  then  abandoned  to  the  action  of  the  forces, 
show  that  the  time  of  a  small  oscillation  in  the  form  of  the 

system  is  27r^f  —  j. 

285.  A  spherical  shell  of  radius  a  and  mass  in  rolls  along  a 
rough  horizontal  plane,  whilst  a  smooth  particle  of  mass  P  oscil- 
lates within  the  shell  in  the  vertical  plane  in  which  the  centre 
of  the  shell  moves,  the  particle  never  being  very  far  from  the 
lowest  point.  Show  that  the  time  of  its  oscillation  will  be  the 
same  as  that  of  a  simple  pendulum  of  length  =■  ina{a^  ■\- Ic^) -^ 
\{in  ^  P^c^ -\- inl^\,  where  k  is  the  radius  of  gyration  of  the 
shell  about  a  diameter. 

286.  A  solid  cylinder  with  projecting  screw-thread  is  freely 
movable  about  its  axis  fixed  vertically,  and  a  hollow  cylinder 
with  a  corresponding  groove  works  freely  about  it  without 
friction.  Find  the  moment  of  the  couple  which  must  act  on 
the  solid  cylinder  in  a  plane  perpendicular  to  its  axis  in  order 
that  the  hollow  cylinder  may  have  no  vertical  motion. 

287.  A  sphere  rolls  from  rest  down  a  given  length  /  of  a 
rough  inclined  plane,  and  then  traverses  a  smooth  part  of  the 
plane  of  length  ml.  Find  the  impulse  which  the  sphere  sustains 
when  perfect  rolling  again  commences,  and  show  that  the  sub- 
sequent velocity  is  less  than  it  would  have  been  if  the  whole 
plane  had  been  rough.  In  the  particular  case  when  m  =  120, 
show  that  the  velocity  is  less  than  it  would  other  vvise  have  been 
in  the  ratio  of  ^J  to  "jy. 

288.  A  rough  sphere  is  placed  upon  a  rough  horizontal  plane 
which  revolves  uniformly  about  a  vertical  axis ;  tlie  centre  of 


MISCELLANEOUS   EXAMPLES. 


233 


the  sphere  is  attracted  to  a  point  in  the  axis  of  rotation,  and  in 
the  same  horizontal  plane  with  itself  by  a  force  varying  as  the 
distance.     Determine  the  motion. 

289.  A  heavy  uniform  beam  AB  is  capable  of  rotating  in  a 
vertical  plane  about  a  fixed  axis  passing  through  its  middle 
point  C,  and  is  inclined  to  the  vertical  at  an  angle  of  60°.  If 
a  perfectly  elastic  ball  fall  upon  it  from  a  given  height,  find 
how  long  a  time  will  elapse  before  the  ball  strikes  the  beam 
again. 

290.  A  sphere  rests  on  a  rough  horizontal  plane,  half  its 
weight  being  supported  by  an  elastic  string  attached  to  the 
highest  point  of  the  sphere ;  the  natural  length  of  the  string 
is  equal  to  the  radius  and  the  stretched  length  to  the  diameter 
of  the  sphere.     If  the  sphere  be  slightly  displaced  parallel  to  a 

vertical  plane,  show  that  the  time  of  an  oscillation  is  '^\{-^\ 

291.  A  uniform  heavy  rod,  movable  about  its  middle  point  A, 
has  its  extremities  connected  with  a  point  B  by  elastic  strings, 
the  natural  length  of  each  of  which  is  equal  to  the  length  AB. 
Find  the  period  of  its  small  oscillations. 

292.  A  squirrel  is  in  a  cylindrical  cage  and  oscillating  with  it 
about  its  axis,  which  is  horizontal.  At  the  instant  when  he  is  at 
the  highest  point  of  the  oscillation,  he  leaps  to  the  opposite 
extremity  of  the  diameter  and  arrives  there  at  the  same  instant 
as  the  point  which  he  left.     Determine  his  leap  completely. 

293.  A  perfectly  rough  sphere  rolls  on  the  internal  surface 

of  a  fixed  cone,  whose  axis  is  vertical  and  vertex  downwards. 

Prove  that  the  angular  velocity  about  its  vertical  diameter  is 

always  the  same  and  that  the  projection  on  a  horizontal  plane 

of  the  radius  vector  of  its  centre,  measured  from  the  axis,  sweeps 

out  areas  proportional  to  the  times.     Show  also  that  the  polar 

equation  to  the  projection  on  a  horizontal  plane  of  the  path  of 

the  centre  is 

(C^n  ,  /      ,    -    •   9    N         5  iT  sin  «  cos  a      2  c^t  cos  a 
+  (2  4-  5  sm2  a)n  =  ^■•^     ,,.,, 4 , 


d&^ 


IhP' 


h 


Jn||f  j 


234 


RIGID   DYNAMICS. 


m  ^ 


i"\ 


fi 


ii' 
.1 


where  a  is  the  semi-vertical  angle,  7  the  constant  angular 
velocity  about  the  vertical  diameter,  and  //  is  half  the  area 
swept  out  by  the  radius  vector  in  a  unit  of  time. 

294.  A  thin  circular  disc  is  set  rotating  on  a  smooth  horizontal 
table,  about  a  vertical  axis  through  its  centre  perpendicular  to 
its  plane,  with  angular  velocity  w  in  a  wind  blowing  with  uniform 
horizontal  velocity  v.  Supposing  the  frictional  resistance  on  a 
small  surface  a  at  rest  to  be  cvma,  where  in  is  the  mass  of  a  unit 
of  area,  show  that  the  angle  turned  through  in  any  time   is 

-  (i  —e""),  and  that  the  centre  of  gravity  moves  through  a  space 
c 

vt — {i—e''*y  Determine  the  same  quantities  for  a  frictional 
resistance  =  cv^ma. 

295.  A  uniform  rod  of  length  2a  passes  through  a  small  fixed 
ring,  its  upper  end  being  constrained  to  move  in  a  horizontal 
straight  groove.  Show  that  if  the  rod  be  slightly  displaced  from 
the  position  of  equilibrium,  the  length  of  the  isochronous  simple 

pendulum  will  be ^ ^,  where  b  is  the  distance  of  the  ring 

from  the  groove. 


3^ 


296.  A  homogeneous  solid  of  revolution  spins  with  great 
rapidity  about  its  axis  of  figure,  which  is  constrained  to  move  in 
the  meridian.  Prove  that  the  axis  will  oscillate  isochronously, 
and  determine  its  positions  of  stable  and  unstable  equilibrium. 

297.  A  wire  in  the  form  of  the  portion  of  the  curve 

cut  off  by  the  initial  line,  rotates  about  the  origin  with  angular 
velocity  eu.     Show  that  the  tendency  to  break  at  a  point  0  =  —  is 

measured  by  ^^  V2  •  mc^aP'^  where  vi  is  the  mass  of  a  unit  of 
length. 

298.  Show  that  in  every  centrobaric  body  the  central  ellipsoid 
of  inertia  is  a  sphere.     Is  the  converse  of  this  proposition  true  .-* 


MISCELLANEOUS   EXAMPLES. 


235 


299.  A  uniform  sphere  is  placed  in  contact  with  the  exterior 
surface  of  a  perfectly  rough  cone.  Its  centre  is  acted  on  by  a 
force  the  direction  of  which  always  meets  the  axis  of  the  cone 
at  right  angles  and  the  intensity  of  which  varies  inversely  as  the 
cube  of  the  distance  from  that  axis.  Prove  that  if  the  sphere 
be  properly  started  the  path  described  by  its  centre  will  meet 
every  generating  line  of  the  cone  on  which  it  lies  at  the  same 
angle. 

300.  A  sphere  of  radius  a  is  suspended  from  a  fixed  point  by 
a  string  of  length  /  and  is  made  to  rotate  about  a  vertical  axis 
with  an  angular  velocity  «.  Prove  that  if  the  string  make  small 
oscillations  about  its  mean  position,  the  motion  of  the  centre  of 
gravity  will  be  represented  by  a  series  of  terms  of  the  form 
L  cos  {kt-\-M\  where  the  several  values  of  k  are  the  roots  of  the 

equation  {IB  -g)  [b  -  (ak  -  ^  =  ^-^— . 

301.  A  rigid  body  is  attached  to  a  fixed  point  by  a  weightless 
string  of  length  /,  which  is  connected  with  the  body  by  a  socket 
(permitting  the  body  to  rotate  freely  without  twisting  the  string) 
at  a  point  on  its  surface  where  an  axis  through  its  centre  of 
inertia,  about  which  the  radius  of  gyration  is  a  maximum  or  a 
minimum,  =  k,  meets  it.  The  body  is  set  rotating  with  angular 
velocity  eo  about  such  axis  placed  vertically  (the  string,  which  is 
tight,  making  an  angle  a  with  the  vertical),  and  being  then  let 
go,  show  that  it  will  ultimately  revolve  with  uniform  angular 
velocity 

302.  Three  equal  uniform  rods  placed  in  a  straight  line  are 
jointed  to  one  another  by  hinges,  and  move  with  a  velocity  v 
perpendicular  to  their  lengths.  If  the  middle  point  of  the 
middle  rod  become  suddenly  fixed,  show  that  the  extremities  of 

the  other  two  will  meet  in  time  -      ,  a  being  the  length  of  each 

rod. 


1 

i: 

r 

W.: 


;i  1* 


i>  ti 


I 


236 


RIGID   DYNAMICS. 


303.  A  top  in  the  form  of  a  surface  of  revolution,  with  a  cir- 
cular plane  end,  is  set  spinning  on  a  smooth  horizontal  plane 
about  its  axis  of  figure,  winch  is  inclined  at  an  angle  a  to  the 
vertical.    It  is  required  to  determine  the  motion  and  to  show  that 

the  axis  will  begin  to  fall  or  to  rise  according  as   tan  «  >  or  <  -, 

a 

where  6  is  the  radius  of  the  circular  plane  end  perpendicular  to 
the  axis,  and  a  is  the  distance  of  the  centre  of  inertia  from  this 
end. 

304.  A  heavy  uniform  beam  AB  of  length  a  is  capable  of 
freely  turning  about  the  point  A,  which  is  fixed;  the  end  B  is 
suspended  from  a  fixed  point  6"  by  a  fine  inextensible  chain  of 
length  c.  The  system  being  at  rest  is  slightly  disturbed.  Find 
the  time  of  a  small  oscillation,  the  weight  of  the  chain  being 
neglected. 

Examine  the  case  in  which  the  line  ACis  vertical. 

305.  A  perfectly  rough  sphere  of  radius  a  moves  on  the  con- 
cave surface  of  a  vertical  cylinder  of  radius  a  +  d,  and  the  centre 
of  the  sphere  initially  has  a  velocity  v  in  a  horizontal  direction. 
Show  that  the  depth  of  its  centre  below  the  initial  position  after 

a  time  /  is  -^^d^i  —  cos;//),  where  n"^  =  ~^. 

Show  also  that  in  order  that  perfect  rolling  may  be  main- 
tained the  coefficient  of  friction  must  not  be  less  than ^. 

306.  A  heavy  particle  slides  down  the  tube  of  an  Archi- 
median  screw,  which  is  vertical  and  capable  of  turning  about 
its  axis.     Determine  the  motion. 


ip^ 

B 

11  ■    ' 

1 

'■' '  "1 

ijh 

if 

ki 

i 

WORKS   ON    PHYSICS,  ETC 


PUDLISIIED   i;V 


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AIRY.     Works  by  Sir  G.  B.  AiUY,  K.C.15.,  lornu'rly  Astronomer- Royal. 

On  Sound  and  Atmospheric  Vibrations.     With  tiie  Mathematical  K'.ements  of 

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Gravitation.     An  Elementary  Explanation  uf  the  Principal  I'urturbatiuns  in  the 
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ALDIS:  Geometrical  Optics.  An  Elementary  Treatise.  By  W.  Sikad.man  .Xi.dis, 
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GALLATLY:  Examples  in  Elementary  Physics.  Comprising  Statics,  Dynamics, 
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GLAZEBROOK:  Heat  and  Light.  By  R.  T.  Glazkbkook,  M.A.,  E.R.S.  C<7m- 
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LOUDON  and  MCLENNAN :  A  Laboratory  Course  of  Experimental  Physics.  By 
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1 2m(.>. 


I 


A  LABORATORY  MANUAL 


OF 


EXPERIMENTAL    PHYSICS. 

W.  J.  LOUDON  and  J.  C.  McLENNAN, 

Demonstrators  in  Physics,  University  of  Toronto. 
Cloth.    8vo.    pp.  302.    $1.90  net. 


FROM  THE  AUTHORS'  PREFACE. 

At  the  present  day,  when  students  are  required  to  gain  Icnowledge  of  natural  phe- 
nomena l)y  performing  experiments  for  themselves  in  laboratories,  every  teacher  finds 
that  as  his  classes  increase  in  numlier,  some  difficulty  is  experienced  in  providing, 
during  a  limited  time,  ample  instruction  in  the  matter  of  details  and  methods. 

During  the  past  few  years  we  ourselves  have  had  such  difficulties  with  large  classes; 
and  that  is  our  reason  for  the  appearance  of  the  present  work,  which  is  the  natural 
outcome  of  our  experience.  We  know  that  it  will  be  of  service  to  our  own  students, 
and  hope  that  it  will  be  appreciated  by  those  engaged  in  teaching  Experimental 
Physics  elsewhere. 

The  book  contains  a  series  of  elementary  experiments  specially  adapted  for  stu- 
dents who  have  had  but  little  acquaintance  with  higher  mathematical  methods:  these 
are  ai ranged,  as  far  as  possible,  in  order  of  difficulty.  There  is  also  an  advanced 
course  of  experimental  work  in  Acoustics,  Ileat,  and  Electricity  and  Magnetism, 
which  is  intended  for  those  who  have  taken  the  elementary  course. 

The  experiments  in  Acoustics  are  simple,  and  of  such  a  nature  that  the  most  of 
them  can  be  performed  by  beginners  in  the  stuay  of  Physics;  those  in  Ileat,  although 
not  recjuiring  more  than  an  ordinar}'  acquaintance  with  Arithmetic,  are  more  tedious 
and  apt  to  test  the  patience  of  the  experimenter;  while  the  course  in  Electricity  and 
Magnetism  has  been  arranged  to  illustrate  the  fundamental  laws  of  the  mathematical 
theory,  and  involves  a  good  working  knov.ledge  of  tlie  Calculus. 


MACMILLAN   &   CO., 

NEW  YORK:    66    FIFTH   AVENUE. 


